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  • richardmitnick 10:56 am on March 14, 2022 Permalink | Reply
    Tags: "Solving problems with intuition", , Birch and Swinnerton-​Dyer (BSD for short) conjecture is one of the most important open problems in the field of number theory., Euler systems: these systems are very complicated mathematical structures that can be used to prove new cases of this conjecture., Mathematics, Number theory is one of the oldest branches of mathematics., Sarah Zerbes - the first ETH professor of theoretical mathematics, The BSD conjecture is one of the most important open problems in the field of number theory,   

    From The Swiss Federal Institute of Technology in Zürich [ETH Zürich] [Eidgenössische Technische Hochschule Zürich] (CH): “Solving problems with intuition” 

    From The Swiss Federal Institute of Technology in Zürich [ETH Zürich] [Eidgenössische Technische Hochschule Zürich] (CH)

    3.14.22
    Barbara Vonarburg

    1
    Sarah Zerbes is the first ETH professor of theoretical mathematics. She finds her best ideas through intuition. Photo: Alessandro Della Bella/ETH Zürich.

    In her research as a number theorist, Sarah Zerbes focuses on one of the oldest – but also most topical – branches of mathematics. Her work is closely tied to one of the great open mathematical problems, the solution for which carries a prize of one million dollars.

    There are two kinds of researchers in mathematics, Zerbes says: “One kind builds theories and sees the big picture.” The other focuses on a particular problem that needs to be solved. “I’m a problem solver,” says the German-​born 43-​year-old, who last autumn was appointed Professor of Mathematics at ETH Zurich. The problems she deals with relate to one of the most famous and mysterious conjectures in mathematics. It was proposed by two British mathematicians, Bryan Birch and Peter Swinnerton-​Dyer, in 1965, after they had spent many nights conducting numerical experiments on what at the time was the sole computer at the University of Cambridge. “These days,” Zerbes says, “anyone could do those calculations on a laptop.”

    Birch and Swinnerton-​Dyer (BSD for short) were unable to prove their conjecture. In 2000, a foundation listed it as one of seven major mathematical problems whose solution would each be rewarded with one million dollars. “It has to do with a class of equations that are very important in mathematics, as well as for some cryptography applications,” Zerbes says: “They’re called elliptic curves.” The challenge is to find certain solutions for these curves. “The BSD conjecture states that the solutions to these equations are determined by an object that stems, surprisingly, from a completely different area of mathematics,” Zerbes says, “namely functions.” This object is known as a complex analytic L-​function.

    Huge network of new conjectures

    The BSD conjecture is one of the most important open problems in the field of number theory, but it has also opened up a new area of research. There is now an extensive network of other conjectures generalising the BSD conjecture. “Together with my husband, I have proved several new sub-​problems in this network,” Zerbes says. She has been collaborating with her husband, David Loeffler, for many years. He is currently a visiting professor at ETH Zürich, alongside his full professorship at the University of Warwick, UK, and works in the same office in the ETH Zürich Main Building as his wife. “Sharing an office isn’t always easy, as it’s very difficult to separate our personal lives from our work. We have the occasional heated discussion,” Zerbes admits, “but we complement each other very well.”

    Unlike herself, her husband is a theory builder who is interested in the big picture. “He has an enormous library in his head and he can understand and categorise things directly.” She’s less adept at this, she says: “My strength is intuition.” Her best ideas come to her when she simply sits and drinks coffee. “I concentrate, contemplate and wait for inspiration,” she says: “I don’t even need a sheet of paper for it.” Only later does she write down her idea in her notebook or on the board in her office, accompanied by much discussion, erasing and rewriting. “First, you always have to see the overarching structure. Only then can you start working out the details, which often takes years,” she says. That’s also what Zerbes and Loeffler experienced in their work in connection with the BSD conjecture.

    Breakthrough after eight years

    “We’ve spent the past eight years developing new examples of Euler systems,” Zerbes says. Named after the Swiss mathematician Leonhard Euler, these systems are very complicated mathematical structures that can be used to prove new cases of this conjecture. Once the fundamental idea was born, the couple was able to finish the first part of their programme within a few years. “But then we were stuck,” Zerbes says. For years, they made no progress, until they flew to a conference in Princeton, US. “There, a mathematician from Lyon gave a lecture in which he presented a tool that he had developed for something else entirely,” she says, “but it was exactly what we were missing.” Although the two mathematicians realized within minutes that they would now succeed, it still took another four years with a lot of work on details. “We achieved the breakthrough last year,” says Zerbes, before summing up by saying, “We were very lucky.”

    But the million-​dollar prize is still out of reach. While it can be shown that the BSD conjecture does indeed hold under certain conditions, there are some cases that no one currently knows how to solve. “We don’t know either,” Zerbes says. “Also, what we’ve proven isn’t parts of the original conjecture, but parts of a generalization; there are other parts that would require a completely new idea.” So the prize isn’t what motivates their research. “It’s the problem itself that’s so fascinating,” Zerbes says: “how deep it is, how complicated the arguments are that might lead to progress, and how lucky one has to be to make any progress.”

    As a number theorist, she also feels connected with generations of mathematicians. “The ancient Greeks of 2,000 years ago were already studying some of the problems that my colleagues and I are working on now,” Zerbes says. Number theory is one of the oldest branches of mathematics. It mostly deals with such equations as the famous Pythagorean theorem: x2+y2=z2. It asks whether integer or rational number solutions can be found for these equations. In the case of Pythagoras, it is known that there are infinitely many rational numbers that solve the equation and that they describe right triangles having sides of length x, y and z. More complicated equations have been keeping mathematicians busy for centuries, and have led to the development of other topics, such as the BSD conjecture.

    Learning Latin as a living language

    In school, Zerbes wasn’t initially interested in mathematics; she preferred Latin. “This language is incredibly analytical and logical,” she says. This is something that still fascinates her today. “I’m now learning Latin as a modern, spoken language,” she says. It bothered her that they only ever translated word for word in school, and that even after six years of lessons she was still incapable of reading a text fluently. Now she has found an instructor who teaches Latin as a living language. “The lessons are conducted exclusively in Latin, and we have discussions and read the ancient texts, which is really interesting,” she says. Only now does she notice how sarcastic, but also funny, Cicero’s writing was.

    As a schoolgirl, she didn’t have any interest in mathematics until, at age 14, she had an outstanding teacher for half a year. “Before that, I didn’t understand maths at all because everything was always packaged in word problems,” Zerbes says. The new teacher was excellent at explaining mathematical concepts. “He was clear, abstract and precise,” she recalls. Now quite interested, when that teacher was replaced again, she took it upon herself to get some mathematics books from the library. After completing her school-​leaving exams, she applied to study at the world-​famous Cambridge University in England and was accepted. She also obtained her doctorate there. When she was later appointed professor at University College London, she invited the teacher from her time at school to attend her introductory lecture. “He actually came, which made me extremely happy,” Zerbes says. “After all, it was his teaching that made all the difference, because that’s when I really started to enjoy maths.”

    Zerbes has since received multiple awards and is one of the world’s leading experts in number theory. She herself has never had any trouble asserting herself as a woman in a male-​dominated environment, but she knows some women in the field who have been bullied because of their sex. “I generally haven’t had any bad experiences,” she says, adding, “I’ve had to develop a thick skin on account of suffering from loss of hair for 35 years, which probably hasn’t hurt, either.” Or maybe, she says, she has just been lucky.

    Mountaineering and ice climbing

    Moving from England to Switzerland was easy for Zerbes. “ETH Zürich is one of the best universities in the world,” she says proudly. “The working conditions and the students are outstanding.” In addition, some of her family lives in southern Germany, and she and her husband are keen mountaineers. “I’m particularly fond of ice climbing,” Zerbes says, “which I recently did in Scuol, in Lower Engadine.” The couple spends most weekends in the mountains, skiing in winter, “to gain another perspective out in nature,” she says, “because otherwise you do dig yourself into quite a deep hole of mathematical problems.” She works out nearly every day, especially swimming and climbing a lot. “Exercise is important to me, as a counterbalance to research,” she says.

    She also finds reading relaxing. Her website features a long list of books she has enjoyed, including such works as Thomas Mann’s Buddenbrooks and Kazuo Ishiguro’s The Remains of the Day. “There are few good books about mathematics,” Zerbes says. There is only one she recommends: Regarding Roderer by Guillermo Martinez, an Argentine mathematician and novelist. Zerbes isn’t bothered by the fact that mathematics is hardly accessible to the general public. She is also happy to overcome the many difficulties that come with the field. She mentions the very first lecture she attended at Cambridge, in which a professor said that mathematics research is bitter and frustrating most of the time. You’re always struggling against the same problems, which can be very draining emotionally. But then, when something works, the feeling is indescribable. “I think of that often,” she says, “because that’s really how it is.”

    See the full article here .

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    Please help promote STEM in your local schools.

    Stem Education Coalition

    ETH Zurich campus

    The Swiss Federal Institute of Technology in Zürich [ETH Zürich] [Eidgenössische Technische Hochschule Zürich] (CH) is a public research university in the city of Zürich, Switzerland. Founded by the Swiss Federal Government in 1854 with the stated mission to educate engineers and scientists, the school focuses exclusively on science, technology, engineering and mathematics. Like its sister institution The Swiss Federal Institute of Technology in Lausanne [EPFL-École Polytechnique Fédérale de Lausanne](CH) , it is part of The Swiss Federal Institutes of Technology Domain (ETH Domain)) , part of the The Swiss Federal Department of Economic Affairs, Education and Research [EAER][Eidgenössisches Departement für Wirtschaft, Bildung und Forschung] [Département fédéral de l’économie, de la formation et de la recherche] (CH).

    The university is an attractive destination for international students thanks to low tuition fees of 809 CHF per semester, PhD and graduate salaries that are amongst the world’s highest, and a world-class reputation in academia and industry. There are currently 22,200 students from over 120 countries, of which 4,180 are pursuing doctoral degrees. In the 2021 edition of the QS World University Rankings ETH Zürich is ranked 6th in the world and 8th by the Times Higher Education World Rankings 2020. In the 2020 QS World University Rankings by subject it is ranked 4th in the world for engineering and technology (2nd in Europe) and 1st for earth & marine science.

    As of November 2019, 21 Nobel laureates, 2 Fields Medalists, 2 Pritzker Prize winners, and 1 Turing Award winner have been affiliated with the Institute, including Albert Einstein. Other notable alumni include John von Neumann and Santiago Calatrava. It is a founding member of the IDEA League and the International Alliance of Research Universities (IARU) and a member of the CESAER network.

    ETH Zürich was founded on 7 February 1854 by the Swiss Confederation and began giving its first lectures on 16 October 1855 as a polytechnic institute (eidgenössische polytechnische schule) at various sites throughout the city of Zurich. It was initially composed of six faculties: architecture, civil engineering, mechanical engineering, chemistry, forestry, and an integrated department for the fields of mathematics, natural sciences, literature, and social and political sciences.

    It is locally still known as Polytechnikum, or simply as Poly, derived from the original name eidgenössische polytechnische schule, which translates to “federal polytechnic school”.

    ETH Zürich is a federal institute (i.e., under direct administration by the Swiss government), whereas The University of Zürich [Universität Zürich ] (CH) is a cantonal institution. The decision for a new federal university was heavily disputed at the time; the liberals pressed for a “federal university”, while the conservative forces wanted all universities to remain under cantonal control, worried that the liberals would gain more political power than they already had. In the beginning, both universities were co-located in the buildings of the University of Zürich.

    From 1905 to 1908, under the presidency of Jérôme Franel, the course program of ETH Zürich was restructured to that of a real university and ETH Zürich was granted the right to award doctorates. In 1909 the first doctorates were awarded. In 1911, it was given its current name, Eidgenössische Technische Hochschule. In 1924, another reorganization structured the university in 12 departments. However, it now has 16 departments.

    ETH Zürich, EPFL (Swiss Federal Institute of Technology in Lausanne) [École polytechnique fédérale de Lausanne](CH), and four associated research institutes form The Domain of the Swiss Federal Institutes of Technology (ETH Domain) [ETH-Bereich; Domaine des Écoles polytechniques fédérales] (CH) with the aim of collaborating on scientific projects.

    Reputation and ranking

    ETH Zürich is ranked among the top universities in the world. Typically, popular rankings place the institution as the best university in continental Europe and ETH Zürich is consistently ranked among the top 1-5 universities in Europe, and among the top 3-10 best universities of the world.

    Historically, ETH Zürich has achieved its reputation particularly in the fields of chemistry, mathematics and physics. There are 32 Nobel laureates who are associated with ETH Zürich, the most recent of whom is Richard F. Heck, awarded the Nobel Prize in chemistry in 2010. Albert Einstein is perhaps its most famous alumnus.

    In 2018, the QS World University Rankings placed ETH Zürich at 7th overall in the world. In 2015, ETH Zürich was ranked 5th in the world in Engineering, Science and Technology, just behind the Massachusetts Institute of Technology (US), Stanford University (US) and University of Cambridge (UK). In 2015, ETH Zürich also ranked 6th in the world in Natural Sciences, and in 2016 ranked 1st in the world for Earth & Marine Sciences for the second consecutive year.

    In 2016, Times Higher Education World University Rankings ranked ETH Zürich 9th overall in the world and 8th in the world in the field of Engineering & Technology, just behind the Massachusetts Institute of Technology(US), Stanford University(US), California Institute of Technology(US), Princeton University(US), University of Cambridge(UK), Imperial College London(UK) and University of Oxford(UK) .

    In a comparison of Swiss universities by swissUP Ranking and in rankings published by CHE comparing the universities of German-speaking countries, ETH Zürich traditionally is ranked first in natural sciences, computer science and engineering sciences.

    In the survey CHE Excellence Ranking on the quality of Western European graduate school programs in the fields of biology, chemistry, physics and mathematics, ETH Zürich was assessed as one of the three institutions to have excellent programs in all the considered fields, the other two being Imperial College London (UK) and the University of Cambridge (UK), respectively.

     
  • richardmitnick 5:52 pm on February 19, 2022 Permalink | Reply
    Tags: "ESO celebrates International Women in Science (part 2)", , , , , , Mathematics   

    From ESOblog (EU): “ESO celebrates International Women in Science (part 2)” 

    From ESOblog (EU)

    At

    ESO 50 Large

    The European Southern Observatory [La Observatorio Europeo Austral] [Observatoire européen austral][Europäische Südsternwarte](EU)(CL)

    18 February 2022
    People@ESO

    2
    Juliet Hannay is part of the science communications team at ESO. She is a former student of the University of Glasgow (SCT) acquiring a Bachelors and Masters degree in Astronomy and Physics. Juliet found a passion for science outreach and communication through her roles as Outreach Convenor, Vice President and President for the Women in STEM society and specialist editor for the Glasgow Insight into Science and Technology magazine.

    2

    As a continuation of ESO’s celebrations of the International Day of Women and Girls in Science, we present part two of our interviews with women at ESO with roles in STEM (Science, technology, engineering, and mathematics). Our interviewees occupy roles ranging from instrumentation engineer to software developer. Here we highlight their individual career paths, presenting a discussion about the issues facing women in the sector. Our interviewees also reflect on what advice they would give their younger selves.

    3
    Teresa Paneque. Credit: T. Paneque.

    A universal language

    Teresa Paneque, a PhD student at ESO in Garching, shares a love of the beauty of maths: “I always liked learning about everything around me,” she reminisces. “When I had my first physics class in high school I thought it was the most amazing thing ever to be able to use maths as a language to understand and predict nature. I thought astronomy was probably the most interesting and coolest area, because the Universe seemed so big that I would never run out of questions to ask and puzzles to solve.” She now uses these skills to ​​study planet-forming discs around stars and understand how they evolve, and shares her passion for astronomy to a large following on social media.

    4
    Patricia Guajardo, working on one of the Unit Telescopes of the VLT (right).Credit: P. Guajardo.

    For Patricia Guajardo, an Instrumentation Engineer at ESO’s Paranal Observatory, her career in STEM was inspired and encouraged by her family, in particular her cousin. “We talked about maths and I didn’t think it was something difficult, so I followed that path,” she recalls. “I began working in telecommunications, where I acquired experience in working remotely with antennas, and this helped me when I applied for a job at ESO. Here I started working as a Telescope & Instrument Operator, carrying out astronomical observations at night. Now I work in the Paranal Instrumentation Group with other engineers to keep the VLT Interferometer [below] in good shape.” The VLT Interferometer combines the light of up to four telescopes, creating a huge “virtual” telescope that allows astronomers to discern details much smaller than what’s possible with each telescope alone.

    5
    Sandra Castro. Credit: L. Calçada.

    “My journey to where I am now had many turns, ups and downs. I wasn’t always interested in STEM,” shares Sandra Castro, the Head of the Pipeline Systems Group in Garching. She leads a team of developers in charge of the software that astronomers use to convert raw data from the telescopes into science-ready data. “During my teen years I wanted to go to medical school before changing my mind to Electronics Engineering. I ended up joining Physics instead and later got a PhD in Astronomy. However, something was already telling me that Astronomy wasn’t my passion. It took me some time to accept that I cared more about the data than the theory behind it.”

    Sandra agrees that work-life balance can be a major challenge in STEM careers, and believes that “more flexible policies for work-life balance and support by the employer are key to fixing challenges faced by women to balance their personal lives with a demanding career in STEM.”

    What is your favorite thing about your job?

    Women in STEM are frequently met with hard questions around this time, focusing on the negatives of working in this sector. It is however equally as important to reflect on our time in STEM, what we are most proud of and our hopes for the future.

    “I feel very fortunate to have worked in different areas of engineering,” Patricia says. “My favourite part is solving problems, being able to contribute to different areas, and learning new things every day. I’d like to stay in this career path, be an expert in this field, and grow together with new discoveries.”

    Sandra shares this enthusiasm for her job: “I love technology, and software is all about trying new technologies. I like to see the results of my creations quickly, and writing software can partially fulfill that for me. I also like that I never stop learning.”

    6
    Eleonora Sani. Credit: E. Sani.

    “The message I got at home was to never allow someone to tell you ‘this is not for you’,” Eleonora shares. “My grandfather was my very first mentor,” she adds. “He taught me everything he knew about our countryside, the fields and the woods. I was his bee-farmer assistant, his mechanical assistant, and he never discouraged me from a job.”

    Diversity is key

    Achieving gender equality in STEM fields is so much more than just ticking a box. By closing the door to a more diverse workforce, an organisation is actively stunting growth, development and scientific progression. One of the most important aspects of any team is diversity of thought and ideas, and this is enabled by having diverse personnel. Although there have been advancements in policies to retain women and encourage them into higher roles, there is still a lack of women in leadership roles in STEM fields.

    “Challenges can be different depending on the working and cultural environment,” explains Eleonora. She found throughout her career that “for a woman, it takes time to be recognised as a professional: they often have to work harder to reach the same level as male colleagues.”

    Sandra adds to this sentiment and remarks that “another thing to take into consideration is that most committees responsible for appointing people to leadership positions are composed mostly of men, which can result in a bias when selecting candidates.”

    “Being a woman in STEM fields can be challenging, because of the lack of visibility of women leading big research topics,” Teresa notes as she discusses her decision to progress in Physics. “I was lucky to have both my masters and PhD investigations supervised by women. However, we need to make women in STEM visible and promote their work so they can inspire and motivate the next generation.”

    “We need all of the brain power in the world,” she continues. “Everyone is welcome and needed, because looking at a problem with different points of views, different experiences and different tools is what will allow us to solve them. Not allowing women into science for so many years was an enormous loss for humanity, they only used half of the brains in the world. It’s like a machine working at 50%, it makes no sense!”

    “Publish or Perish”

    It is no secret that a career in academia or STEM can be a high stress environment. With long and unusual working hours as well as deadlines for publishing and applying for funding, it can be hard to strike the right balance of professional and personal life.

    “Having a good work-life balance is very difficult,” Eleonora explains. “The academic environment is becoming more and more competitive over time and a stable position comes 5-10 years after completing a PhD, and often you have to move more than once. This leads many valuable researchers to face a difficult and unfair choice: their private life or their career.”

    “An organisation should have policies which support families with flexible working schedules, respecting and promoting a healthy life-work balance,” she concludes.

    Eleonora likes “to face the unexpected, to take decisions under pressure and to investigate issues on the fly. It is particularly rewarding to learn by experience: you may occasionally be wrong, of course, but you always learn a lot from your mistakes!” What about the future? “In the mid term I want to tackle the challenges of bringing the Extremely Large Telescope into operation. In the longer term, let’s say more than ten years or so, my goal is to become a leader recognised outside my own area of research. Who knows where this can lead, maybe far, maybe I will change country or institute. One of my personal statements is ‘never say never’.”

    Based on these experiences and those of other ESO women in part 1 of this series, it is clear that this is a multi-faceted problem. Besides encouraging women into STEM, it is also key to implement policies that retain them in the field. This includes reviewing recruitment and career development procedures, or minimising systematic biases in the allocation of resources like telescope observing time, which ESO does via gender-balanced panels and double-blind evaluations. To promote a welcoming working environment for all, and especially for women, ESO is working on a Diversity, Equity and Inclusion Plan to address these issues. After all, the issues faced by women in STEM reach far beyond a single day, and so too should our efforts!

    See the full article here .


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    European Southern Observatory [La Observatorio Europeo Austral] [Observatoire européen austral][Europäische Südsternwarte] (EU) is the foremost intergovernmental astronomy organisation in Europe and the world’s most productive ground-based astronomical observatory by far. It is supported by 16 countries: Austria, Belgium, Brazil, the Czech Republic, Denmark, France, Finland, Germany, Italy, the Netherlands, Poland, Portugal, Spain, Sweden, Switzerland and the United Kingdom, along with the host state of Chile. ESO carries out an ambitious programme focused on the design, construction and operation of powerful ground-based observing facilities enabling astronomers to make important scientific discoveries. ESO also plays a leading role in promoting and organising cooperation in astronomical research. ESO operates three unique world-class observing sites in Chile: Cerro La Silla, Paranal and Chajnantor. At Paranal, ESO operates the Very Large Telescope, the world’s most advanced visible-light astronomical observatory and two survey telescopes. VISTA works in the infrared and is the world’s largest survey telescope and the VLT Survey Telescope is the largest telescope designed to exclusively survey the skies in visible light. ESO is a major partner in ALMA, the largest astronomical project in existence. And on Cerro Armazones, close to Paranal, ESO is building the 39-metre European Extremely Large Telescope, the E-ELT, which will become “the world’s biggest eye on the sky”.

    European Southern Observatory(EU) La Silla HELIOS (HARPS Experiment for Light Integrated Over the Sun).

    ESO 3.6m telescope & HARPS at Cerro LaSilla, Chile, 600 km north of Santiago de Chile at an altitude of 2400 metres.

    MPG Institute for Astronomy [Max-Planck-Institut für Astronomie](DE) 2.2 meter telescope at/European Southern Observatory(EU) Cerro La Silla, Chile, 600 km north of Santiago de Chile at an altitude of 2400 metres.

    European Southern Observatory(EU) La Silla Observatory 600 km north of Santiago de Chile at an altitude of 2400 metres.

    European Southern Observatory(EU) , Very Large Telescope at Cerro Paranal in the Atacama Desert •ANTU (UT1; The Sun ) •KUEYEN (UT2; The Moon ) •MELIPAL (UT3; The Southern Cross ), and •YEPUN (UT4; Venus – as evening star). Elevation 2,635 m (8,645 ft) from above Credit J.L. Dauvergne & G. Hüdepohl atacama photo.

    European Southern Observatory(EU)VLTI Interferometer image, Cerro Paranal, with an elevation of 2,635 metres (8,645 ft) above sea level, •ANTU (UT1; The Sun ),•KUEYEN (UT2; The Moon ),•MELIPAL (UT3; The Southern Cross ), and •YEPUN (UT4; Venus – as evening.

    ESO VLT Survey telescope.

    ESO Very Large Telescope 4 lasers on Yepun (CL)

    Glistening against the awesome backdrop of the night sky above ESO’s Paranal Observatory, four laser beams project out into the darkness from Unit Telescope 4 UT4 of the VLT, a major asset of the Adaptive Optics system.

    ESO New Technology Telescope at Cerro La Silla , Chile, at an altitude of 2400 metres.

    Part of ESO’s Paranal Observatory, the VLT Survey Telescope (VISTA) observes the brilliantly clear skies above the Atacama Desert of Chile. It is the largest survey telescope in the world in visible light, with an elevation of 2,635 metres (8,645 ft) above sea level.

    European Southern Observatory/National Radio Astronomy Observatory(US)/National Astronomical Observatory of Japan(JP) ALMA Array in Chile in the Atacama at Chajnantor plateau, at 5,000 metres.

    European Southern Observatory(EU) ELT 39 meter telescope to be on top of Cerro Armazones in the Atacama Desert of northern Chile. located at the summit of the mountain at an altitude of 3,060 metres (10,040 ft).

    European Southern Observatory(EU)/MPG Institute for Radio Astronomy [MPG Institut für Radioastronomie](DE) ESO’s Atacama Pathfinder Experiment(CL) high on the Chajnantor plateau in Chile’s Atacama region, at an altitude of over 4,800 m (15,700 ft).

    The Leiden Observatory [Sterrewacht Leiden](NL) MASCARA instrument cabinet at Cerro La Silla, located in the southern Atacama Desert 600 kilometres (370 mi) north of Santiago de Chile at an altitude of 2,400 metres (7,900 ft).

    ESO Next Generation Transit Survey telescopes, an array of twelve robotic 20-centimetre telescopes at Cerro Paranal,(CL) 2,635 metres (8,645 ft) above sea level.

    ESO Speculoos telescopes four 1 meter robotic telescopes at ESO Paranal Observatory 2635 metres 8645 ft above sea level.

    TAROT telescope at Cerro LaSilla, 2,635 metres (8,645 ft) above sea level.

    European Southern Observatory (EU) ExTrA telescopes at Cerro LaSilla at an altitude of 2400 metres.

    A novel gamma ray telescope under construction on Mount Hopkins, Arizona. A large project known as the Čerenkov Telescope Array composed of hundreds of similar telescopes to be situated in the Canary Islands and Chile at, ESO Cerro Paranal site The telescope on Mount Hopkins will be fitted with a prototype high-speed camera, assembled at the. University of Wisconsin–Madison and capable of taking pictures at a billion frames per second. Credit: Vladimir Vassiliev.

    European Space Agency [La Agencia Espacial Europea] [Agence spatiale européenne][Europäische Weltraumorganisation](EU), The new Test-Bed Telescope 2 is housed inside the shiny white dome shown in this picture, at ESO’s LaSilla Facility in Chile. The telescope has now started operations and will assist its northern-hemisphere twin in protecting us from potentially hazardous, near-Earth objects.The domes of ESO’s 0.5 m and the Danish 0.5 m telescopes are visible in the background of this image.

    Part of the world-wide effort to scan and identify near-Earth objects, the European Space Agency’s Test-Bed Telescope 2 (TBT2), a technology demonstrator hosted at ESO’s La Silla Observatory in Chile, has now started operating. Working alongside its northern-hemisphere partner telescope, TBT2 will keep a close eye on the sky for asteroids that could pose a risk to Earth, testing hardware and software for a future telescope network.

    European Space Agency [La Agencia Espacial Europea] [Agence spatiale européenne][Europäische Weltraumorganisation](EU) The open dome of The black telescope structure of the‘s Test-Bed Telescope 2 peers out of its open dome in front of the rolling desert landscape. The telescope is located at ESO’s La Silla Observatory, which sits at a 2400 metre altitude in the Chilean Atacama desert.

     
  • richardmitnick 11:10 am on February 10, 2022 Permalink | Reply
    Tags: "Imaginary Numbers Are Reality", , , Imaginary numbers are not imaginary at all. The truth is they have had far more impact on our lives than anything truly imaginary ever could., Mathematics, , The square roots of negative numbers are what we now call imaginary numbers., Without imaginary numbers and the vital role they played in putting electricity into homes and factories and internet server-farms the modern world would not exist.   

    From Nautilus (US): “Imaginary Numbers Are Reality” 

    February 9, 2022
    Michael Brooks

    1

    Imaginary numbers are not imaginary at all. The truth is they have had far more impact on our lives than anything truly imaginary ever could. Without imaginary numbers and the vital role they played in putting electricity into homes and factories and internet server-farms the modern world would not exist. Students who might complain to their math teacher that there’s no point in anyone learning how to use imaginary numbers would have to put down their phone, turn off their music, and pull the wires out of their broadband router. But perhaps we should start with an explanation of what an imaginary number is.

    We know by now how to square a number (multiply it by itself), and we know that negative numbers make a positive number when squared; a minus times a minus is a plus, remember? So (–2) × (–2) = 4. We also know that taking a square root is the inverse of squaring. So the possible square roots of 4 are 2 and –2. The imaginary number arises from asking what the square root of –4 would be.

    Surely the question is meaningless? If you square a number, whether positive or negative, the answer is positive. So you can’t do the inverse operation if you start with a negative number. That’s certainly what Heron of Alexandria seemed to think. Heron was the Egyptian architect whose mathematical tricks, written in Stereometrica, gave us the dome of the Hagia Sophia. In the same volume, he showed how to calculate the volume of a truncated square pyramid; that is, a pyramid with the top chopped off. His solution for one example involved subtracting 288 from 225 and finding the square root of the result. The result, though, is a negative number: –63. So the answer would be found via √–63.

    For some reason—whether a sense that there was some mistake, or someone copied something down wrong, or because it was so absurd—the manuscripts we have show that Heron ignored the minus sign and gave the answer as √63 instead.

    The square roots of negative numbers are what we now call imaginary numbers. The first person to suggest that they shouldn’t be ignored was the 16th-century Italian astrologer Jerome Cardano, who was embarked on a grand project: a book detailing all of the algebraic knowledge of his times. While working out cubic equations, he stopped and stared at the issue. At first, he called them “impossible cases.” In his 1545 book on algebra, The Great Art, he gave the example of trying to divide 10 into two numbers that multiply together to make 40. In the process of finding those numbers, you come across 5 + √–15.

    Cardano didn’t shy away from this unexpected encounter. In fact, he even jotted down a few thoughts about it. However, he wrote in Latin, and translators argue about what he actually meant. For some, he calls it a “false position.” For others, it’s a “fictitious” number. Still others say he characterizes the situation as “impossible” to solve. One of his further comments on how to proceed in such a situation is translated as “putting aside the mental tortures” and as “the imaginary parts being lost.” Elsewhere he refers to this as “arithmetic subtlety, the end of which … is as refined as it is useless.” He says it “truly is sophisticated … one cannot carry out the other operations one can in the case of a pure negative.” By pure negative, he means a standard negative number, something like –4. He was happy with negative numbers and wrote that “√9 is either +3 or –3, for a plus [times a plus] or a minus times a minus yields a plus.” And then he continued, “√–9 is neither +3 or –3 but is some recondite third sort of thing.” Cardano clearly thought the square roots of negative numbers were something abstruse and abstract, but at the same time he knew they were something—and something that a mathematician should engage with. The task wasn’t for him, though; none of Cardano’s subsequent writings mention the square roots of negative numbers. He left it to his fellow countryman, Rafael Bombelli, to address them a couple of decades or so later.

    In what he called a “wild thought,” Bombelli suggested in 1572 that the two terms in 5 + √–15 could be treated as two separate things. “The whole matter seemed to rest on sophistry rather than truth,” he said, but he did it anyway. And we still do it today because it works.

    Bombelli’s two separate things were what we now call real numbers and imaginary numbers. The combination of the two is known as a “complex number” (it’s complex as in “military-industrial complex,” speaking of combination—of real and imaginary parts—rather than complication). But let’s be clear. If there’s one thing we’ve learned in our time revisiting mathematics, it’s that all numbers are imaginary. They are simply a notation that helps with the concept of “how many.” So applying the name “imaginary numbers” to the square roots of negative numbers is pejorative and unhelpful.

    That said, we should acknowledge a distinction. What mathematicians call “real” numbers are the numbers you’re more familiar with. The “two” in two apples; the 3.14… in pi; the fraction. And just as positive numbers are in a sense complemented by negative numbers, what we call real numbers are complemented by what we now have to call imaginary numbers. Think of them as yin and yang, or heads and tails. And certainly not as actually imaginary.

    Bombelli, in his wild thought, demonstrated that this new tribe of numbers have a role to play in the real world. He set out to solve a cubic equation that Cardano had given up on: x3 = 15x + 4. Cardano’s solution required him to deal with an expression that contained the square root of –121, and he just didn’t know where to go with it. Bombelli, on the other hand, thought he might try applying normal rules of arithmetic to the square root. So, he said, maybe √–121 is the same as √121 × √–1, which gives 11 × √–1.

    Bombelli’s great breakthrough was to see that these strange, seemingly impossible numbers obey simple arithmetic rules once they are separated out from the other, more familiar types of number during a calculation. Everything after that was just grasping the nettle.

    Proceeding with Cardano’s cubic equation, he eventually arrived at a solution:

    Separate them out into what we would now call their real and imaginary parts, and it simplifies to 2 plus 2, and √–1 minus √–1. The imaginary part disappears, leaving us with just 2 + 2. So x = 4 is one of the solutions to x3 = 15x + 4. Plug it in and check for yourself.

    These days, the convention is to use i to represent √–1. The Swiss mathematician Leonard Euler first came up with this. It’s easy to assume that i stands for imaginary, but the truth is, as with his e, Euler may just have picked it at random. Whatever the reason, Euler’s move has cemented i as the imaginary number in a very unhelpful way.

    To see better what an imaginary number is, let’s think of a standard number line that runs from –1 to 1 (you can think of it as a ruler placed on a table in front of you, running from –1 on the left to +1 on the right). We call the process of moving along the line addition and subtraction (I’m at 0.3, and I’ll add 0.3 more, which takes me to 0.6). But we can also imagine making some moves by multiplication. If I start at 1, how do I get to –1? I multiply by –1. So let’s picture multiplication by –1 as half a rotation, anticlockwise, around a circle (in our case, the circle passes through 1 and –1). It’s actually a rotation by 180 degrees. In mathematicians’ preferred units to denote angles, 180 degrees is π radians (360°, a whole circle, is 2π radians).

    2

    What happens if we only do half of this rotation? It’s halfway to multiplying by –1, which you can think of as the same as multiplying by √–1. That rotation, by just π/2 radians (or 90°) leaves our number up on the top part of the circle’s circumference, away from the standard number line. So we can think of the square root of –1 as sitting on a number line that runs at right angles to the number line we’re familiar with. It’s just another set of numbers, this time on a ruler that meets your other ruler at 90° to form a cross, with +1 at the end furthest from you, and –1 right in front of you.

    That leads us somewhere interesting. The link with rotation in circles means that i is related to π and the sines and cosines of angles. That relationship is mediated through the strange number e, often called Euler’s number. This “irrational” number begins with the sequence 2.71828… and goes on forever. It is ubiquitous in mathematics and is vital to statistics, calculus, natural logarithms, and a range of arithmetic calculations. Euler worked out exactly what this looks like by taking a particular kind of infinite series (it’s called a Taylor series), and deriving something now known as Euler’s formula:

    e±iθ = cosθ ± isinθ

    This shows there is a fundamental relationship between the base of the natural logarithm and the imaginary number. What’s more, you can reduce this to the relation known as the Euler identity:

    eiπ + 1 = 0

    To some, this is a near-mystical formula. Here we have the base of natural logarithms e; the numbers 0 and 1, which are both unique cases on the whole number line; the imaginary number, a special case all of its own; and π, which as we know is a source of power in mathematics. Despite being discovered at different times by different people looking at different pieces of mathematics, it turns out they are interrelated, coexisting in this elegant, simple equation.

    Seen from a slightly different perspective, perhaps we shouldn’t be surprised. As with π itself, there really isn’t anything mystical about this formula. It results from the fact that numbers change and transform themselves and each other through rotations. That only happens because of what numbers are: representations of the relationships between quantities. We don’t find anything mystical about moving along the familiar “real” number line by adding and subtracting. And there’s nothing different, really, about the transformations that come about through multiplications and divisions. Remember that sines and cosines are just ratios—one number divided by another—that are related to the angles within triangles, and you can represent those angles as fractions or multiples of π in units known as radians. So what we’re discovering here is not some deep mystery about the universe, but a clear and useful set of relationships that are a consequence of defining numbers in various different ways.

    In fact, these relationships are more than useful—they could be described as vital. Take their application to science, for example: a full mathematical description of nature seems to require imaginary numbers to exist. The “real” numbers, of which we have learned so much, are not enough. They must be combined with the imaginary numbers to form the “complex” numbers that Bombelli first created. The result, says mathematician Roger Penrose, is a beautiful completeness. “Complex numbers, as much as reals, and perhaps even more, find a unity with nature that is truly remarkable,” he says in his book The Road to Reality. “It is as though Nature herself is as impressed by the scope and consistency of the complex-number system as we are ourselves, and has entrusted to these numbers the precise operations of her world at its minutest scales.” In other words, imaginary numbers had to be discovered because they are an essential part of the description of nature.

    See the full article here .

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    Welcome to Nautilus (US). We are delighted you joined us. We are here to tell you about science and its endless connections to our lives. Each month we choose a single topic. And each Thursday we publish a new chapter on that topic online. Each issue combines the sciences, culture and philosophy into a single story told by the world’s leading thinkers and writers. We follow the story wherever it leads us. Read our essays, investigative reports, and blogs. Fiction, too. Take in our games, videos, and graphic stories. Stop in for a minute, or an hour. Nautilus lets science spill over its usual borders. We are science, connected.

     
  • richardmitnick 9:45 am on January 24, 2022 Permalink | Reply
    Tags: "At the interface of physics and mathematics", , , Integrable model: equation that can be solved exactly., Mathematics, , , , String Theory-which scientists hope will eventually provide a unified description of particle physics and gravity., ,   

    From The Swiss Federal Institute of Technology in Zürich [ETH Zürich] [Eidgenössische Technische Hochschule Zürich] (CH): “At the interface of physics and mathematics” 

    From The Swiss Federal Institute of Technology in Zürich [ETH Zürich] [Eidgenössische Technische Hochschule Zürich] (CH)

    24.01.2022
    Barbara Vonarburg

    Sylvain Lacroix is a theoretical physicist who conducts research into fundamental concepts of physics – an exciting but intellectually challenging field of science. As an Advanced Fellow at ETH Zürich’s Institute for Theoretical Studies (ITS), he works on complex equations that can be solved exactly only thanks to their large number of symmetries.

    1
    “It was fascinating to learn abstract mathematical concepts and see them neatly applied in the realm of physics,” says Sylvain Lacroix, Advanced Fellow at the Institute for Theoretical Studies. Photo: Nicola Pitaro/ETH Zürich.

    “I got hooked on the interplay of physics and mathematics while I was still at secondary school,” says 30-​year-old Sylvain Lacroix, who was born and grew up near Paris. “It was fascinating to learn abstract mathematical concepts and see them neatly applied in the realm of physics.” During his studies at The University of Lyon [Université Claude Bernard Lyon 1] (FR), he devoted much of his energy and enthusiasm to physics problems that had highly complex underlying mathematical structures. So when it came to selecting a topic for his doctoral thesis, this area of research seemed like the obvious choice. He decided to explore the theory of what are known as integrable models – a subject he has continued to pursue up to the present day.

    Lacroix readily acknowledges that most people outside his line of work find the term “integrable models” completely incomprehensible: “I have to admit that it’s probably not the simplest or most accessible field of physics,” he says, almost apologetically. That’s why he takes pains to explain it in layman’s terms: “We define a model as a body of laws, a set of equations that describe the behaviour of certain quantities, for example how the position of an object changes over time.” An integrable model is characterised by equations that can be solved exactly, which is by no means a given.

    Symmetry is the key

    Many of the equations used in modern physics – such as that practised at The European Organization for Nuclear Research [Organización Europea para la Investigación Nuclear][Organisation européenne pour la recherche nucléaire] [Europäische Organisation für Kernforschung](CH) [CERN] – are so complex that they can be solved only approximately. These approximation methods often serve their purpose well, for instance if there is only a weak interaction between two particles. However, other cases require exact calculations – and that’s where integrable models come in. But what makes them so exact? “That’s another aspect that is tricky to explain,” Lacroix says, “but it ultimately comes down to symmetry.” Take, for example, the symmetry of time or space: a physics experiment will produce the same results whether you perform it today or – under identical conditions – ten days from now, and whether it takes place in Zürich or New York. Consequently, the equation that describes the experiment must remain invariant even if the time or location changes. This is reflected in the mathematical structure of the equation, which contains the corresponding constraints. “If we have enough symmetries, this results in so many constraints that we can simplify the equation to the point where we get exact results,” says the physicist.

    Integrable models and their exact solutions are actually very rare in mathematics. “If I chose a random equation, it would be extremely unlikely to have this property of exact solvability,” Lacroix says. “But equations of this kind really do exist in nature.” Some describe the movement of waves propagating in a channel, for example, while others describe the behaviour of a hydrogen atom. “But it’s important to note that my work doesn’t have any practical applications of that kind,” Lacroix says. “I don’t examine concrete physical models; instead, I study mathematical structures and attempt to find general approaches that will allow us to construct new exactly solvable equations.” Although some of these formulas may eventually find a real-​world application, others probably won’t.

    After completing his doctoral thesis, Lacroix spent three years working as a postdoc at The University of Hamburg [Universität Hamburg](DE), before finally moving to Zürich in September 2021. “I don’t have a family, so I had no problem making the switch,” he says. He is relieved that he can now spend five years at the ITS as an Advanced Fellow and focus entirely on his research without having to worry about the future. He admits it was a pleasure getting to know different countries as a postdoc and that he enjoyed moving from place to place. “But it makes it very hard to have any kind of stability in your life.”

    A beautiful setting

    Lacroix spends most of his time working in his office at the ITS, which is located in a stately building dating from 1882 not far from the ETH Main Building. “It’s a lovely place,” he says, glancing out the window at the green surroundings and the city beyond. “I feel very much at home here. Living in Zürich is wonderful, it’s such a great feeling being here.” In his spare time, he likes watching movies, reading books and socialising. “I love meeting up with friends in restaurants or cafés,” he says. He also feels fortunate that he didn’t start working in Zürich until after the Covid measures had been relaxed.

    “I’m vaccinated and everyone’s very careful at ETH. We still have restrictions in place, but life is slowly getting back to normal – and that made it much easier to get to know my colleagues from day one,” he says. One of the greatest privileges of working at the ITS, Lacroix says, is that it offers an international environment that brings together researchers from all over the world. As well as offering a space for experts to exchange ideas and holding seminars where Fellows can present their work, the Institute also has a tradition of organising joint excursions. In the autumn of 2021, Lacroix joined his colleagues on a hike in the Flumserberg mountain resort for the first time: “I love hiking and it’s incredible to have the mountains so close.”

    Normally, however, he can be found sitting at his desk jotting down a series of mostly abstract equations on a sheet of paper. Occasionally his computer comes in handy, he says, because it has become so much more than just a calculating device; today’s computers can also handle abstract mathematical concepts, which can be very useful. Most people don’t really understand much of what Lacroix puts down on paper, but that doesn’t bother him: “I’ve learned to live with that,” he says; “I don’t feel isolated in my research at all – at least not in the academic sphere.”

    A better understanding of quantum field theory

    Integrable models are extremely symmetrical models, Lacroix explains. The basic principle of symmetry plays an important role in modern physics, for example in quantum field theory – the theoretical basis of particle physics – as well as in string theory, which scientists hope will eventually provide a unified description of particle physics and gravity. So could such an all-​encompassing unified field theory turn out to be an integrable model? “That would obviously be great, especially for me!” Lacroix says with a wry smile. “But it’s a bit optimistic to believe that whatever unified theory of physics finally emerges will have enough symmetries to make it completely exact.”

    Even if the equations he studies don’t explain the world directly, he still believes they can help us achieve a better understanding of theoretical physics. For example, we can take advantage of so-​called “toy models”, which have a particularly large number of symmetries, to simplify extremely complex equations in quantum field theory. “This gives us a better understanding of how the theory works, even if these models are too simplistic for the real world,” Lacroix says. Yet his primary interest lies in the purely mathematical questions that integrable models pose, and he admits that the equations they involve sometimes even appear in his dreams: “It’s hard to shake off what I’ve been thinking about the entire day. But I’ve never managed to solve a mathematical problem in my dreams – at least not so far!”

    See the full article here .

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    Please help promote STEM in your local schools.

    Stem Education Coalition

    ETH Zurich campus

    The Swiss Federal Institute of Technology in Zürich [ETH Zürich] [Eidgenössische Technische Hochschule Zürich] (CH) is a public research university in the city of Zürich, Switzerland. Founded by the Swiss Federal Government in 1854 with the stated mission to educate engineers and scientists, the school focuses exclusively on science, technology, engineering and mathematics. Like its sister institution The Swiss Federal Institute of Technology in Lausanne [EPFL-École Polytechnique Fédérale de Lausanne](CH) , it is part of The Swiss Federal Institutes of Technology Domain (ETH Domain)) , part of the The Swiss Federal Department of Economic Affairs, Education and Research [EAER][Eidgenössisches Departement für Wirtschaft, Bildung und Forschung] [Département fédéral de l’économie, de la formation et de la recherche] (CH).

    The university is an attractive destination for international students thanks to low tuition fees of 809 CHF per semester, PhD and graduate salaries that are amongst the world’s highest, and a world-class reputation in academia and industry. There are currently 22,200 students from over 120 countries, of which 4,180 are pursuing doctoral degrees. In the 2021 edition of the QS World University Rankings ETH Zürich is ranked 6th in the world and 8th by the Times Higher Education World Rankings 2020. In the 2020 QS World University Rankings by subject it is ranked 4th in the world for engineering and technology (2nd in Europe) and 1st for earth & marine science.

    As of November 2019, 21 Nobel laureates, 2 Fields Medalists, 2 Pritzker Prize winners, and 1 Turing Award winner have been affiliated with the Institute, including Albert Einstein. Other notable alumni include John von Neumann and Santiago Calatrava. It is a founding member of the IDEA League and the International Alliance of Research Universities (IARU) and a member of the CESAER network.

    ETH Zürich was founded on 7 February 1854 by the Swiss Confederation and began giving its first lectures on 16 October 1855 as a polytechnic institute (eidgenössische polytechnische Schule) at various sites throughout the city of Zurich. It was initially composed of six faculties: architecture, civil engineering, mechanical engineering, chemistry, forestry, and an integrated department for the fields of mathematics, natural sciences, literature, and social and political sciences.

    It is locally still known as Polytechnikum, or simply as Poly, derived from the original name eidgenössische polytechnische Schule, which translates to “federal polytechnic school”.

    ETH Zürich is a federal institute (i.e., under direct administration by the Swiss government), whereas The University of Zürich [Universität Zürich ] (CH) is a cantonal institution. The decision for a new federal university was heavily disputed at the time; the liberals pressed for a “federal university”, while the conservative forces wanted all universities to remain under cantonal control, worried that the liberals would gain more political power than they already had. In the beginning, both universities were co-located in the buildings of the University of Zürich.

    From 1905 to 1908, under the presidency of Jérôme Franel, the course program of ETH Zürich was restructured to that of a real university and ETH Zürich was granted the right to award doctorates. In 1909 the first doctorates were awarded. In 1911, it was given its current name, Eidgenössische Technische Hochschule. In 1924, another reorganization structured the university in 12 departments. However, it now has 16 departments.

    ETH Zürich, EPFL (Swiss Federal Institute of Technology in Lausanne) [École polytechnique fédérale de Lausanne](CH), and four associated research institutes form The Domain of the Swiss Federal Institutes of Technology (ETH Domain) [ETH-Bereich; Domaine des Écoles polytechniques fédérales] (CH) with the aim of collaborating on scientific projects.

    Reputation and ranking

    ETH Zürich is ranked among the top universities in the world. Typically, popular rankings place the institution as the best university in continental Europe and ETH Zürich is consistently ranked among the top 1-5 universities in Europe, and among the top 3-10 best universities of the world.

    Historically, ETH Zürich has achieved its reputation particularly in the fields of chemistry, mathematics and physics. There are 32 Nobel laureates who are associated with ETH Zürich, the most recent of whom is Richard F. Heck, awarded the Nobel Prize in chemistry in 2010. Albert Einstein is perhaps its most famous alumnus.

    In 2018, the QS World University Rankings placed ETH Zürich at 7th overall in the world. In 2015, ETH Zürich was ranked 5th in the world in Engineering, Science and Technology, just behind the Massachusetts Institute of Technology(US), Stanford University(US) and University of Cambridge(UK). In 2015, ETH Zürich also ranked 6th in the world in Natural Sciences, and in 2016 ranked 1st in the world for Earth & Marine Sciences for the second consecutive year.

    In 2016, Times Higher Education World University Rankings ranked ETH Zürich 9th overall in the world and 8th in the world in the field of Engineering & Technology, just behind the Massachusetts Institute of Technology(US), Stanford University(US), California Institute of Technology(US), Princeton University(US), University of Cambridge(UK), Imperial College London(UK) and University of Oxford(UK) .

    In a comparison of Swiss universities by swissUP Ranking and in rankings published by CHE comparing the universities of German-speaking countries, ETH Zürich traditionally is ranked first in natural sciences, computer science and engineering sciences.

    In the survey CHE ExcellenceRanking on the quality of Western European graduate school programs in the fields of biology, chemistry, physics and mathematics, ETH Zürich was assessed as one of the three institutions to have excellent programs in all the considered fields, the other two being Imperial College London(UK) and The University of Cambridge(UK), respectively.

     
  • richardmitnick 5:04 pm on January 3, 2022 Permalink | Reply
    Tags: "Liber Abaci" first introduced the sequence to the Western world., "What is the Fibonacci sequence?", Ancient Sanskrit texts that used the Hindu-Arabic numeral system first mention it in 200 B.C. predating Leonardo of Pisa by centuries., Fibonacci sequence can be described using a mathematical equation: Xn+2= Xn+1 + Xn, From its origins to its significance almost every popular notion about the famous Fibonacci sequence is wrong., In 1202 Leonardo of Pisa published the massive tome "Liber Abaci" a mathematics "cookbook" for how to do calculations., It would take a large book to document all the misinformation about the golden ratio much of which is simply the repetition of the same errors by different authors., It's true that the Fibonacci sequence is tightly connected to what's now known as the golden ratio-phi- an irrational number that has a great deal of its own dubious lore., Leonardo of Pisa did not actually discover the sequence., , Mathematics, Other than being a neat teaching tool the Fibonacci sequence shows up in a few places in nature., The Fibonacci sequence is a series of numbers in which each number is the sum of the two that precede it. Starting at 0 and 1 the sequence looks like this: 0; 1; 1; 2; 3; 5; 8; 13; 21; 34 & foreve, The first thing to know is that the sequence is not originally Fibonacci's, The golden ratio manages to capture some types of plant growth., The Italian mathematician who we call Leonardo Fibonacci was born around 1170 and originally known as Leonardo of Pisa, The ratio of successive numbers in the Fibonacci sequence gets ever closer to the golden ratio which is 1.6180339887498948482..., The sequence is not some secret code that governs the architecture of the universe., The sequence was mostly forgotten until the 19th century when mathematicians worked out more about the sequence's mathematical properties.   

    From Live Science: “What is the Fibonacci sequence?” 

    From Live Science

    1.3.22
    Tia Ghose

    From its origins to its significance almost every popular notion about the famous Fibonacci sequence is wrong.

    1
    The seeds in a sunflower exhibit a golden spiral, which is tied to the Fibonacci sequence. Image credit: belterz/Getty Images.

    The Fibonacci sequence is a series of numbers in which each number is the sum of the two that precede it. Starting at 0 and 1, the sequence looks like this: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so on forever. Fibonacci sequence can be described using a mathematical equation: Xn+2= Xn+1 + Xn

    People claim there are many special properties about the numerical sequence, such as the fact that it is “nature’s secret code” for building perfect structures, like the Great Pyramid at Giza or the iconic seashell that likely graced the cover of your school mathematics textbook. But much of that is incorrect and the true history of the series is a bit more down-to-earth.

    The first thing to know is that the sequence is not originally Fibonacci’s, who in fact never went by that name. The Italian mathematician who we call Leonardo Fibonacci was born around 1170 and originally known as Leonardo of Pisa, said Keith Devlin, a mathematician at Stanford University (US).

    Only in the 19th century did historians come up with the nickname Fibonacci (roughly meaning, “son of the Bonacci clan”), to distinguish the mathematician from another famous Leonardo of Pisa, Devlin said.

    Leonardo of Pisa did not actually discover the sequence, said Devlin, who is also the author of Finding Fibonacci: The Quest to Rediscover the Forgotten Mathematical Genius Who Changed the World (Princeton University Press, 2017). Ancient Sanskrit texts that used the Hindu-Arabic numeral system first mention it in 200 B.C. predating Leonardo of Pisa by centuries.

    “It’s been around forever,” Devlin told Live Science.

    However, in 1202 Leonardo of Pisa published the massive tome Liber Abaci, a mathematics “cookbook for how to do calculations,” Devlin said. Written for tradesmen, Liber Abaci laid out Hindu-Arabic arithmetic useful for tracking profits; losses; remaining loan balances and so on he added.

    In one place in the book, Leonardo of Pisa introduces the sequence with a problem involving rabbits. The problem goes as follows: Start with a male and a female rabbit. After a month, they mature and produce a litter with another male and female rabbit. A month later, those rabbits reproduce and out comes — you guessed it — another male and female, who also can mate after a month. (Ignore the wildly improbable biology here.) After a year, how many rabbits would you have?

    The answer, it turns out, is 144 — and the formula used to get to that answer is what’s now known as the Fibonacci sequence.

    Liber Abaci first introduced the sequence to the Western world. But after a few scant paragraphs on breeding rabbits, Leonardo of Pisa never mentioned the sequence again. In fact, it was mostly forgotten until the 19th century when mathematicians worked out more about the sequence’s mathematical properties. In 1877, French mathematician Édouard Lucas officially named the rabbit problem “the Fibonacci sequence,” Devlin said.

    2
    The Fibonacci sequence and the golden ratio are eloquent equations, but they aren’t as magical as they may seem. (Image credit: Shutterstock)

    Other than being a neat teaching tool the Fibonacci sequence shows up in a few places in nature. However, it’s not some secret code that governs the architecture of the universe, Devlin said.

    It’s true that the Fibonacci sequence is tightly connected to what’s now known as the golden ratio, phi, an irrational number that has a great deal of its own dubious lore. The ratio of successive numbers in the Fibonacci sequence gets ever closer to the golden ratio which is 1.6180339887498948482…

    The golden ratio manages to capture some types of plant growth, Devlin said. For instance, the spiral arrangement of leaves or petals on some plants follows the golden ratio. Pinecones exhibit a golden spiral, as do the seeds in a sunflower, according to Phyllotaxis: A Systemic Study in Plant Morphogenesis (Cambridge University Press, 1994). But there are just as many plants that do not follow this rule.

    “It’s not ‘God’s only rule’ for growing things, let’s put it that way,” Devlin said.

    Perhaps the most famous example of all, the seashell known as the nautilus, does not in fact grow new cells according to the Fibonacci sequence, he added. When people start to draw connections to the human body, art and architecture, links to the Fibonacci sequence go from tenuous to downright fictional.

    “It would take a large book to document all the misinformation about the golden ratio much of which is simply the repetition of the same errors by different authors,” George Markowsky, a mathematician who was then at The University of Maine (US), wrote in a 1992 paper in the College Mathematics Journal.

    Much of this misinformation can be attributed to an 1855 book by the German psychologist Adolf Zeising called Aesthetic Research. Zeising claimed the proportions of the human body were based on the golden ratio. In subsequent years, the golden ratio sprouted “golden rectangles,” “golden triangles” and all sorts of theories about where these iconic dimensions crop up.

    Since then, people have said the golden ratio can be found in the dimensions of the Pyramid at Giza, the Parthenon, Leonardo da Vinci’s Vitruvian Man and a bevy of Renaissance buildings. Overarching claims about the ratio being “uniquely pleasing” to the human eye have been stated uncritically, Devlin said. All these claims, when they’re tested, are measurably false, he added.

    “We’re good pattern recognizers. We can see a pattern regardless of whether it’s there or not,” Devlin said. “It’s all just wishful thinking.”

    See the full article here .

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  • richardmitnick 1:56 pm on January 3, 2022 Permalink | Reply
    Tags: "Mathematicians Outwit Hidden Number Conspiracy", A new proof has debunked a conspiracy that mathematicians feared might haunt the number line., An improved solution to a particular formulation of the Chowla conjecture., , , Chowla conjecture, Connected numbers represent exceptions to Chowla’s conjecture in which the factorization of one integer actually does bias that of another., Consider the number 1001 which is divisible by the primes 7; 11; and 13. In Tao’s graph it shares edges with 1008; 1012 and 1014 (by addition) as well as with 994; 990 and 988 (by subtraction)., Eigenvalues, Expander graphs, Expander graphs have previously led to new discoveries in theoretical computer science; group theory and other areas of math., Harald Helfgott of The University of Göttingen [Georg-August-Universität Göttingen](DE), Helfgott and Radziwiłł have expander graphs available for problems in number theory as well., Helfgott and Radziwiłł’s solution to the logarithmic Chowla conjecture marked a significant quantitative improvement on Tao’s result., Linking two arithmetic operations that usually live independently of one another., Liouville function, Maksym Radziwiłł of The California Institute of Technology (US), Many of number theory’s most important problems arise when mathematicians think about how multiplication and addition relate in terms of the prime numbers., Mathematics, Problems about primes that involve addition have plagued mathematicians for centuries., Proving the Chowla conjecture is a “sort of warmup or steppingstone” to answering those more intractable problems., , Tao proved an easier version of the problem called the logarithmic Chowla conjecture., Terence Tao of The University of California-Los Angeles (US), The primes themselves are defined in terms of multiplication: They’re divisible by no numbers other than themselves and 1., The twin primes conjecture asserts that there are infinitely many primes that differ by only 2 (like 11 and 13)., There could be this vast conspiracy that every time a number n decides to be prime it has some secret agreement with its neighbor n + 2 saying you’re not allowed to be prime anymore., This work has given mathematicians another set of tools for understanding arithmetic’s fundamental building blocks-the prime numbers., When multiplied together they construct the rest of the integers.   

    From Quanta Magazine (US) : “Mathematicians Outwit Hidden Number Conspiracy” 

    From Quanta Magazine (US)

    January 3, 2022
    Jordana Cepelewicz

    A new proof has debunked a conspiracy that mathematicians feared might haunt the number line. In doing so, it has given them another set of tools for understanding arithmetic’s fundamental building blocks-the prime numbers.

    In a paper posted last March, Harald Helfgott of The University of Göttingen [Georg-August-Universität Göttingen](DE) and Maksym Radziwiłł of The California Institute of Technology (US) presented an improved solution to a particular formulation of the Chowla conjecture, a question about the relationships between integers.

    The conjecture predicts that whether one integer has an even or odd number of prime factors does not influence whether the next or previous integer also has an even or odd number of prime factors. That is, nearby numbers do not collude about some of their most basic arithmetic properties.

    That seemingly straightforward inquiry is intertwined with some of math’s deepest unsolved questions about the primes themselves. Proving the Chowla conjecture is a “sort of warmup or steppingstone” to answering those more intractable problems, said Terence Tao of The University of California-Los Angeles (US).

    3
    Terence Tao developed a strategy for using expander graphs to answer a version of the Chowla conjecture but couldn’t quite make it work. Courtesy of UCLA.

    And yet for decades, that warmup was a nearly impossible task itself. It was only a few years ago that mathematicians made any progress, when Tao proved an easier version of the problem called the logarithmic Chowla conjecture. But while the technique he used was heralded as innovative and exciting, it yielded a result that was not precise enough to help make additional headway on related problems, including ones about the primes. Mathematicians hoped for a stronger and more widely applicable proof instead.

    Now, Helfgott and Radziwiłł have provided just that. Their solution, which pushes techniques from graph theory squarely into the heart of number theory, has reignited hope that the Chowla conjecture will deliver on its promise — ultimately leading mathematicians to the ideas they’ll need to confront some of their most elusive questions.

    Conspiracy Theories

    Many of number theory’s most important problems arise when mathematicians think about how multiplication and addition relate in terms of the prime numbers.

    The primes themselves are defined in terms of multiplication: They’re divisible by no numbers other than themselves and 1, and when multiplied together they construct the rest of the integers. But problems about primes that involve addition have plagued mathematicians for centuries. For instance, the twin primes conjecture asserts that there are infinitely many primes that differ by only 2 (like 11 and 13). The question is challenging because it links two arithmetic operations that usually live independently of one another. “It’s difficult because we are mixing two worlds,” said Oleksiy Klurman of The University of Bristol (UK).

    1
    Maksym Radziwiłł. Caltech.

    2
    Harald Helfgott . University of Göttingen.

    Intuition tells mathematicians that adding 2 to a number should completely change its multiplicative structure — meaning there should be no correlation between whether a number is prime (a multiplicative property) and whether the number two units away is prime (an additive property). Number theorists have found no evidence to suggest that such a correlation exists, but without a proof, they can’t exclude the possibility that one might emerge eventually.

    “For all we know, there could be this vast conspiracy that every time a number n decides to be prime, it has some secret agreement with its neighbor n + 2 saying you’re not allowed to be prime anymore,” said Tao.

    No one has come close to ruling out such a conspiracy. That’s why, in 1965, Sarvadaman Chowla formulated a slightly easier way to think about the relationship between nearby numbers. He wanted to show that whether an integer has an even or odd number of prime factors — a condition known as the “parity” of its number of prime factors — should not in any way bias the number of prime factors of its neighbors.

    This statement is often understood in terms of the Liouville function, which assigns integers a value of −1 if they have an odd number of prime factors (like 12, which is equal to 2 × 2 × 3) and +1 if they have an even number (like 10, which is equal to 2 × 5). The conjecture predicts that there should be no correlation between the values that the Liouville function takes for consecutive numbers.

    Many state-of-the-art methods for studying prime numbers break down when it comes to measuring parity, which is precisely what Chowla’s conjecture is all about. Mathematicians hoped that by solving it, they’d develop ideas they could apply to problems like the twin primes conjecture.

    For years, though, it remained no more than that: a fanciful hope. Then, in 2015, everything changed.

    Dispersing Clusters

    Radziwiłł and Kaisa Matomäki of The University of Turku [Turun yliopisto](FI) didn’t set out to solve the Chowla conjecture. Instead, they wanted to study the behavior of the Liouville function over short intervals. They already knew that, on average, the function is +1 half the time and −1 half the time. But it was still possible that its values might cluster, cropping up in long concentrations of either all +1s or all −1s.

    In 2015, Matomäki and Radziwiłł proved that those clusters almost never occur [Annals of Mathematics]. Their work, published the following year, established that if you choose a random number and look at, say, its hundred or thousand nearest neighbors, roughly half have an even number of prime factors and half an odd number.

    “That was the big piece that was missing from the puzzle,” said Andrew Granville of The University of Montreal [Université de Montréal](CA). “They made this unbelievable breakthrough that revolutionized the whole subject.”

    It was strong evidence that numbers aren’t complicit in a large-scale conspiracy — but the Chowla conjecture is about conspiracies at the finest level. That’s where Tao came in. Within months, he saw a way to build on Matomäki and Radziwiłł’s work to attack a version of the problem that’s easier to study, the logarithmic Chowla conjecture. In this formulation, smaller numbers are given larger weights so that they are just as likely to be sampled as larger integers.

    Tao had a vision for how a proof of the logarithmic Chowla conjecture might go. First, he would assume that the logarithmic Chowla conjecture is false — that there is in fact a conspiracy between the number of prime factors of consecutive integers. Then he’d try to demonstrate that such a conspiracy could be amplified: An exception to the Chowla conjecture would mean not just a conspiracy among consecutive integers, but a much larger conspiracy along entire swaths of the number line.

    He would then be able to take advantage of Radziwiłł and Matomäki’s earlier result, which had ruled out larger conspiracies of exactly this kind. A counterexample to the Chowla conjecture would imply a logical contradiction — meaning it could not exist, and the conjecture had to be true.

    But before Tao could do any of that, he had to come up with a new way of linking numbers.

    A Web of Lies

    Tao started by capitalizing on a defining feature of the Liouville function. Consider the numbers 2 and 3. Both have an odd number of prime factors and therefore share a Liouville value of −1. But because the Liouville function is multiplicative, multiples of 2 and 3 also have the same sign pattern as each other.

    That simple fact carries an important implication. If 2 and 3 both have an odd number of prime factors due to some secret conspiracy, then there’s also a conspiracy between 4 and 6 — numbers that differ not by 1 but by 2. And it gets worse from there: A conspiracy between adjacent integers would also imply conspiracies between all pairs of their multiples.

    “For any prime, these conspiracies will propagate,” Tao said.

    To better understand this widening conspiracy, Tao thought about it in terms of a graph — a collection of vertices connected by edges. In this graph, each vertex represents an integer. If two numbers differ by a prime and are also divisible by that prime, they’re connected by an edge.

    For example consider the number 1001, which is divisible by the primes 7, 11 and 13. In Tao’s graph, it shares edges with 1,008, 1,012 and 1,014 (by addition), as well as with 994, 990 and 988 (by subtraction). Each of these numbers is in turn connected to many other vertices.

    4
    Samuel Velasco/Quanta Magazine

    Taken together, those edges encode broader networks of influence: Connected numbers represent exceptions to Chowla’s conjecture in which the factorization of one integer actually does bias that of another.

    To prove his logarithmic version of the Chowla conjecture, Tao needed to show that this graph has too many connections to be a realistic representation of values of the Liouville function. In the language of graph theory, that meant showing that his graph of interconnected numbers had a specific property — that it was an “expander” graph.

    Expander Walks

    An expander is an ideal yardstick for measuring the scope of a conspiracy. It’s a highly connected graph, even though it has relatively few edges compared to its number of vertices. That makes it difficult to create a cluster of interconnected vertices that don’t interact much with other parts of the graph.

    If Tao could show that his graph was a local expander — that any given neighborhood on the graph had this property — he’d prove that a single breach of the Chowla conjecture would spread across the number line, a clear violation of Matomäki and Radziwiłł’s 2015 result.

    “The only way to have correlations is if the entire population sort of shares that correlation,” said Tao.

    Proving that a graph is an expander often translates to studying random walks along its edges. In a random walk, each successive step is determined by chance, as if you were wandering through a city and flipping a coin at each intersection to decide whether to turn left or right. If the streets of that city form an expander, it’s possible to get pretty much anywhere by taking random walks of relatively few steps.

    But walks on Tao’s graph are strange and circuitous. It’s impossible, for instance, to jump directly from 1,001 to 1,002; that requires at least three steps. A random walk along this graph starts at an integer, adds or subtracts a random prime that divides it, and moves to another integer.

    It’s not obvious that repeating this process only a few times can lead to any point in a given neighborhood, which should be the case if the graph really is an expander. In fact, when the integers on the graph get big enough, it’s no longer clear how to even create random paths: Breaking numbers down into their prime factors — and therefore defining the graph’s edges — becomes prohibitively difficult.

    “It’s a scary thing, counting all these walks,” Helfgott said.

    When Tao tried to show that his graph was an expander, “it was a little too hard,” he said. He developed a new approach instead, based on a measure of randomness called entropy. This allowed him to circumvent the need to show the expander property — but at a cost.

    He could solve the logarithmic Chowla conjecture [Forum of Mathematics, Pi], but less precisely than he’d wanted to. In an ideal proof of the conjecture, independence between integers should always be evident, even along small sections of the number line. But with Tao’s proof, that independence doesn’t become visible until you sample over an astronomical number of integers.

    “It’s not quantitatively very strong,” said Joni Teräväinen of the University of Turku.

    Moreover, it wasn’t clear how to extend his entropy method to other problems.

    “Tao’s work was a complete breakthrough,” said James Maynard of The University of Oxford (UK), but because of those limitations, “it couldn’t possibly give those things that would lead to the natural next steps in the direction of problems more like the twin primes conjecture.”

    Five years later, Helfgott and Radziwiłł managed to do what Tao couldn’t — by extending the conspiracy he’d identified even further.

    Enhancing the Conspiracy

    Tao had built a graph that connected two integers if they differed by a prime and were divisible by that prime. Helfgott and Radziwiłł considered a new, “naïve” graph that did away with that second condition, connecting numbers merely if subtracting one from the other yielded a prime.

    The effect was an explosion of edges. On this naïve graph, 1,001 didn’t have just six connections with other vertices, it had hundreds. But the graph was also much simpler than Tao’s in a key way: Taking random walks along its edges didn’t require knowledge of the prime divisors of very large integers. That, along with the greater density of edges, made it much easier to demonstrate that any neighborhood in the naïve graph had the expander property — that you’re likely to get from any vertex to any other in a small number of random steps.

    Helfgott and Radziwiłł needed to show that this naïve graph approximated Tao’s graph. If they could show that the two graphs were similar, they would be able to infer properties of Tao’s graph by looking at theirs instead. And because they already knew their graph was a local expander, they’d be able to conclude that Tao’s was, too (and therefore that the logarithmic Chowla conjecture was true).

    But given that the naïve graph had so many more edges than Tao’s, the resemblance was buried, if it existed at all.

    “What does it even mean when you’re saying these graphs look like each other?” Helfgott said.

    Hidden Resemblance

    While the graphs don’t look like each other on the surface, Helfgott and Radziwiłł set out to prove that they approximate each other by translating between two perspectives. In one, they looked at the graphs as graphs; in the other, they looked at them as objects called matrices.

    First they represented each graph as a matrix, which is an array of values that in this case encoded connections between vertices. Then they subtracted the matrix that represented the naïve graph from the matrix that represented Tao’s graph. The result was a matrix that represented the difference between the two.

    Helfgott and Radziwiłł needed to prove that certain parameters associated with this matrix, called eigenvalues, were all small. This is because a defining characteristic of an expander graph is that its associated matrix has one large eigenvalue while the rest are significantly smaller. If Tao’s graph, like the naïve one, was an expander, then it too would have one large eigenvalue — and those two large eigenvalues would nearly cancel out when one matrix was subtracted from the other, leaving a set of eigenvalues that were all small.

    But eigenvalues are tricky to study by themselves. Instead, an equivalent way to prove that all the eigenvalues of this matrix were small involved a return to graph theory. And so, Helfgott and Radziwiłł converted this matrix (the difference between the matrices representing their naïve graph and Tao’s more complicated one) back into a graph itself.

    They then proved that this graph contained few random walks — of a certain length and in compliance with a handful of other properties — that looped back to their starting points. This implied that most random walks on Tao’s graph had essentially canceled out random walks on the naïve expander graph — meaning that the former could be approximated by the latter, and both were therefore expanders.

    A Way Forward

    Helfgott and Radziwiłł’s solution to the logarithmic Chowla conjecture marked a significant quantitative improvement on Tao’s result. They could sample over far fewer integers to arrive at the same outcome: The parity of the number of prime factors of an integer is not correlated with that of its neighbors.

    “That’s a very strong statement about how prime numbers and divisibility look random,” said Ben Green of Oxford.

    But the work is perhaps even more exciting because it provides “a natural way to attack the problem,” Matomäki said — exactly the intuitive approach that Tao first hoped for six years ago.

    Expander graphs have previously led to new discoveries in theoretical computer science; group theory and other areas of math. Now, Helfgott and Radziwiłł have made them available for problems in number theory as well. Their work demonstrates that expander graphs have the power to reveal some of the most basic properties of arithmetic — dispelling potential conspiracies and starting to disentangle the complex interplay between addition and multiplication.

    “Suddenly, when you’re using the graph language, it’s seeing all this structure in the problem that you couldn’t really see beforehand,” Maynard said. “That’s the magic.”

    See the full article here .


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    Formerly known as Simons Science News, Quanta Magazine (US) is an editorially independent online publication launched by the Simons Foundation to enhance public understanding of science. Why Quanta? Albert Einstein called photons “quanta of light.” Our goal is to “illuminate science.” At Quanta Magazine, scientific accuracy is every bit as important as telling a good story. All of our articles are meticulously researched, reported, edited, copy-edited and fact-checked.

     
  • richardmitnick 9:48 am on September 20, 2021 Permalink | Reply
    Tags: "Taking on the stormy seas", Mathematics, ,   

    From The Massachusetts Institute of Technology (US) : “Taking on the stormy seas” 

    MIT News

    From The Massachusetts Institute of Technology (US)

    September 19, 2021
    Michaela Jarvis

    1
    Themistoklis Sapsis, an associate professor of mechanical engineering at MIT, uses analytical and computational methods to try to predict behavior — such as that of ocean waves or instability inside a gas turbine — amid uncertain and occasionally extreme dynamics. Credit: M. Scott Brauer.

    On his first day of classes at the Technical University of Athens’ School of Naval Architecture and Marine Engineering, Themistoklis Sapsis had a very satisfying realization.

    “I realized that ships and other maritime structures are the only ones that operate at the interface of two different media: air and water,” says Sapsis. “This property alone creates so many challenges in terms of mathematical and computational modeling. And, of course, these media are not calm at all — they are random and often surprisingly unpredictable.”

    In other words, Sapsis did not have to choose between his two great passions: huge, ocean-going ships and structures on the one hand, and mathematics on the other. Today, Sapsis, an associate professor of mechanical engineering at MIT, uses analytical and computational methods to try to predict behavior — such as that of ocean waves or instability inside a gas turbine — amid uncertain and occasionally extreme dynamics. His goal is to create designs for structures that are robust and safe even in a broad range of conditions. For example, he may study the loads acting on a ship during a storm, or the flow separation and lift reduction around a helicopter rotor blade during a difficult maneuver.

    “These events are real — they often lead to big catastrophes and casualties,” Sapsis says. “My goal is to predict them and develop algorithms that can simulate them quickly. If we achieve this goal, then we could start talking about optimization and design of these systems with consideration of these extreme, rare, but possibly catastrophic events.”

    Growing up in Athens, where great seafaring and mathematical traditions date back to ancient times, Sapsis’ house was “full of machine elements, spare engines, and engineering blueprints,” the tools of his father’s trade as a superintendent engineer in the maritime industry.

    His father traveled internationally to oversee major ship repairs, and Sapsis often went along.

    “I think what made the biggest impression on me as a child was the size of these vessels and especially the engines. You had to climb five or six flights of stairs to see the whole thing,” he recalls.

    Also in the Sapsis home were math and engineering books — “lots of them,” he says. His father insisted that he study math closely, at the same time that the young Sapsis was conducting physics experiments in the basement.

    “This back-and-forth transition between dynamical systems — more generally mathematics — and naval architecture” was frequently on his mind, Sapsis says.

    In college, Sapsis ended up taking every math class that was offered. He says he had the good fortune to get in touch early on with the most mathematically inclined professor in the School of Naval Architecture and Marine Engineering, who then mentored Sapsis for three years. In his spare time, Sapsis even attended classes in the university’s School of Applied Mathematics.

    His undergraduate thesis was on probabilistic description of dynamical systems subjected to random excitations, a topic important to the understanding of the motions of large ships and loads. One of Sapsis’ most memorable research breakthroughs occurred while he was working on that thesis.

    “I was given a nice problem by my thesis advisor,” Sapsis says. “He warned me that most likely I would not be able to get something new, as this was an old problem and many had tried in the past decades without success.”

    Over the next six months, Sapsis went over every step of the methods that were in the academic literature, “again and again,” he says, trying to understand why various approaches failed. He started to discern a path toward deriving a new set of equations that could achieve his goal, but there were technical obstacles.

    “Without a lot of hope, as I knew that his was an old problem, but with a lot of curiosity, I began working on the different steps,” Sapsis says. “After a few weeks of work, I realized that the steps were complete, and I had a new set of equations!”

    “It was certainly one of my most enthusiastic moments,” Sapsis says, “when I heard my advisor saying, ‘Yes, this is new and it is important!’”

    Since that early success, the engineering and architecture problems associated with building for the extreme and unpredictable ocean environment have provided Sapsis with plenty of research problems to solve.

    “Naval architecture is one of the oldest professions, with many open problems remaining and many more new ones coming,” he says. “The theoretical tools should not be more complex than the problem itself. However, in this case there are some really challenging physical problems that require the development of fundamentally new mathematics and computational methods. I am always trying to begin with the fundamentals and build the right theoretical and computational tools to, hopefully, come closer to the modeling of certain complex phenomena.”

    Sapsis, who joined the MIT faculty in 2013 and was tenured in 2019, says he loves the energy and pace of the Institute, where “there are so many things happening here that you can never feel you have achieved enough — but in a healthy way.”

    “I always feel humbled by the amazing achievements of my colleagues and our students and postdocs,” he says. “It is a place filled with pure passion and talent, blended together for a good cause, to solve the world’s hardest problems.”

    These days, Sapsis says it is his students who experience the pure excitement of finding solutions to problems in the field.

    “My students and postdocs are now the ones who have the pleasure to be the first to find out when a new idea works,” Sapsis says. “I have to admit, however, that I save some problems for myself.”

    In fact, Sapsis says he relaxes by “thinking about a nice problem: a high-risk and low-expectations one. I think of a strategy to go about it but know that most likely it will not work. This is something I don’t consider work.”

    See the full article here .


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    MIT Seal

    USPS “Forever” postage stamps celebrating Innovation at MIT.

    MIT Campus

    The Massachusetts Institute of Technology (US) is a private land-grant research university in Cambridge, Massachusetts. The institute has an urban campus that extends more than a mile (1.6 km) alongside the Charles River. The institute also encompasses a number of major off-campus facilities such as the The MIT Lincoln Laboratory (US), The MIT Bates Research and Engineering Center (US), and The Haystack Observatory (US), as well as affiliated laboratories such as The Broad Institute of MIT and Harvard(US) and The Whitehead Institute (US).

    Founded in 1861 in response to the increasing industrialization of the United States, The Massachusetts Institute of Technology adopted a European polytechnic university model and stressed laboratory instruction in applied science and engineering. It has since played a key role in the development of many aspects of modern science, engineering, mathematics, and technology, and is widely known for its innovation and academic strength. It is frequently regarded as one of the most prestigious universities in the world.

    As of December 2020, 97 Nobel laureates, 26 Turing Award winners, and 8 Fields Medalists have been affiliated with MIT as alumni, faculty members, or researchers. In addition, 58 National Medal of Science recipients, 29 National Medals of Technology and Innovation recipients, 50 MacArthur Fellows, 80 Marshall Scholars, 3 Mitchell Scholars, 22 Schwarzman Scholars, 41 astronauts, and 16 Chief Scientists of the U.S. Air Force have been affiliated with Massachusetts Institute of Technology (US) . The university also has a strong entrepreneurial culture and MIT alumni have founded or co-founded many notable companies. The Massachusetts Institute of Technology is a member of the Association of American Universities (AAU).

    Foundation and vision

    In 1859, a proposal was submitted to the Massachusetts General Court to use newly filled lands in Back Bay, Boston for a “Conservatory of Art and Science”, but the proposal failed. A charter for the incorporation of the Massachusetts Institute of Technology, proposed by William Barton Rogers, was signed by John Albion Andrew, the governor of Massachusetts, on April 10, 1861.

    Rogers, a professor from the University of Virginia (US), wanted to establish an institution to address rapid scientific and technological advances. He did not wish to found a professional school, but a combination with elements of both professional and liberal education, proposing that:

    “The true and only practicable object of a polytechnic school is, as I conceive, the teaching, not of the minute details and manipulations of the arts, which can be done only in the workshop, but the inculcation of those scientific principles which form the basis and explanation of them, and along with this, a full and methodical review of all their leading processes and operations in connection with physical laws.”

    The Rogers Plan reflected the German research university model, emphasizing an independent faculty engaged in research, as well as instruction oriented around seminars and laboratories.

    Early developments

    Two days after The Massachusetts Institute of Technology was chartered, the first battle of the Civil War broke out. After a long delay through the war years, MIT’s first classes were held in the Mercantile Building in Boston in 1865. The new institute was founded as part of the Morrill Land-Grant Colleges Act to fund institutions “to promote the liberal and practical education of the industrial classes” and was a land-grant school. In 1863 under the same act, the Commonwealth of Massachusetts founded the Massachusetts Agricultural College, which developed as The University of Massachusetts-Amherst (US). In 1866, the proceeds from land sales went toward new buildings in the Back Bay.

    The Massachusetts Institute of Technology was informally called “Boston Tech”. The institute adopted the European polytechnic university model and emphasized laboratory instruction from an early date. Despite chronic financial problems, the institute saw growth in the last two decades of the 19th century under President Francis Amasa Walker. Programs in electrical, chemical, marine, and sanitary engineering were introduced, new buildings were built, and the size of the student body increased to more than one thousand.

    The curriculum drifted to a vocational emphasis, with less focus on theoretical science. The fledgling school still suffered from chronic financial shortages which diverted the attention of the MIT leadership. During these “Boston Tech” years, The Massachusetts Institute of Technology faculty and alumni rebuffed Harvard University (US) president (and former MIT faculty) Charles W. Eliot’s repeated attempts to merge MIT with Harvard College’s Lawrence Scientific School. There would be at least six attempts to absorb MIT into Harvard. In its cramped Back Bay location, MIT could not afford to expand its overcrowded facilities, driving a desperate search for a new campus and funding. Eventually, the MIT Corporation approved a formal agreement to merge with Harvard, over the vehement objections of MIT faculty, students, and alumni. However, a 1917 decision by the Massachusetts Supreme Judicial Court effectively put an end to the merger scheme.

    In 1916, The Massachusetts Institute of Technology administration and the MIT charter crossed the Charles River on the ceremonial barge Bucentaur built for the occasion, to signify MIT’s move to a spacious new campus largely consisting of filled land on a one-mile-long (1.6 km) tract along the Cambridge side of the Charles River. The neoclassical “New Technology” campus was designed by William W. Bosworth and had been funded largely by anonymous donations from a mysterious “Mr. Smith”, starting in 1912. In January 1920, the donor was revealed to be the industrialist George Eastman of Rochester, New York, who had invented methods of film production and processing, and founded Eastman Kodak. Between 1912 and 1920, Eastman donated $20 million ($236.6 million in 2015 dollars) in cash and Kodak stock to MIT.

    Curricular reforms

    In the 1930s, President Karl Taylor Compton and Vice-President (effectively Provost) Vannevar Bush emphasized the importance of pure sciences like physics and chemistry and reduced the vocational practice required in shops and drafting studios. The Compton reforms “renewed confidence in the ability of the Institute to develop leadership in science as well as in engineering”. Unlike Ivy League schools, The Massachusetts Institute of Technology catered more to middle-class families, and depended more on tuition than on endowments or grants for its funding. The school was elected to the Association of American Universities (US)in 1934.

    Still, as late as 1949, the Lewis Committee lamented in its report on the state of education at The Massachusetts Institute of Technology that “the Institute is widely conceived as basically a vocational school”, a “partly unjustified” perception the committee sought to change. The report comprehensively reviewed the undergraduate curriculum, recommended offering a broader education, and warned against letting engineering and government-sponsored research detract from the sciences and humanities. The School of Humanities, Arts, and Social Sciences and The MIT Sloan School of Management were formed in 1950 to compete with the powerful Schools of Science and Engineering. Previously marginalized faculties in the areas of economics, management, political science, and linguistics emerged into cohesive and assertive departments by attracting respected professors and launching competitive graduate programs. The School of Humanities, Arts, and Social Sciences continued to develop under the successive terms of the more humanistically oriented presidents Howard W. Johnson and Jerome Wiesner between 1966 and 1980.

    The Massachusetts Institute of Technology‘s involvement in military science surged during World War II. In 1941, Vannevar Bush was appointed head of the federal Office of Scientific Research and Development and directed funding to only a select group of universities, including MIT. Engineers and scientists from across the country gathered at The Massachusetts Institute of Technology’s Radiation Laboratory, established in 1940 to assist the British military in developing microwave radar. The work done there significantly affected both the war and subsequent research in the area. Other defense projects included gyroscope-based and other complex control systems for gunsight, bombsight, and inertial navigation under Charles Stark Draper’s Instrumentation Laboratory; the development of a digital computer for flight simulations under Project Whirlwind; and high-speed and high-altitude photography under Harold Edgerton. By the end of the war, The Massachusetts Institute of Technology became the nation’s largest wartime R&D contractor (attracting some criticism of Bush), employing nearly 4000 in the Radiation Laboratory alone and receiving in excess of $100 million ($1.2 billion in 2015 dollars) before 1946. Work on defense projects continued even after then. Post-war government-sponsored research at MIT included SAGE and guidance systems for ballistic missiles and Project Apollo.

    These activities affected The Massachusetts Institute of Technology profoundly. A 1949 report noted the lack of “any great slackening in the pace of life at the Institute” to match the return to peacetime, remembering the “academic tranquility of the prewar years”, though acknowledging the significant contributions of military research to the increased emphasis on graduate education and rapid growth of personnel and facilities. The faculty doubled and the graduate student body quintupled during the terms of Karl Taylor Compton, president of The Massachusetts Institute of Technology between 1930 and 1948; James Rhyne Killian, president from 1948 to 1957; and Julius Adams Stratton, chancellor from 1952 to 1957, whose institution-building strategies shaped the expanding university. By the 1950s, The Massachusetts Institute of Technology no longer simply benefited the industries with which it had worked for three decades, and it had developed closer working relationships with new patrons, philanthropic foundations and the federal government.

    In late 1960s and early 1970s, student and faculty activists protested against the Vietnam War and The Massachusetts Institute of Technology’s defense research. In this period The Massachusetts Institute of Technology’s various departments were researching helicopters, smart bombs and counterinsurgency techniques for the war in Vietnam as well as guidance systems for nuclear missiles. The Union of Concerned Scientists was founded on March 4, 1969 during a meeting of faculty members and students seeking to shift the emphasis on military research toward environmental and social problems. The Massachusetts Institute of Technology ultimately divested itself from the Instrumentation Laboratory and moved all classified research off-campus to the The MIT Lincoln Laboratory facility in 1973 in response to protests. The student body, faculty, and administration remained comparatively unpolarized during what was a tumultuous time for many other universities. Johnson was seen to be highly successful in leading his institution to “greater strength and unity” after these times of turmoil. However six Massachusetts Institute of Technology students were sentenced to prison terms at this time and some former student leaders, such as Michael Albert and George Katsiaficas, are still indignant about MIT’s role in military research and its suppression of these protests. (Richard Leacock’s film, November Actions, records some of these tumultuous events.)

    In the 1980s, there was more controversy at The Massachusetts Institute of Technology over its involvement in SDI (space weaponry) and CBW (chemical and biological warfare) research. More recently, The Massachusetts Institute of Technology’s research for the military has included work on robots, drones and ‘battle suits’.

    Recent history

    The Massachusetts Institute of Technology has kept pace with and helped to advance the digital age. In addition to developing the predecessors to modern computing and networking technologies, students, staff, and faculty members at Project MAC, The Artificial Intelligence Laboratory, and The Tech Model Railroad Club wrote some of the earliest interactive computer video games like Spacewar! and created much of modern hacker slang and culture. Several major computer-related organizations have originated at MIT since the 1980s: Richard Stallman’s GNU Project and the subsequent Free Software Foundation were founded in the mid-1980s at the AI Lab; The MIT Media Lab was founded in 1985 by Nicholas Negroponte and Jerome Wiesner to promote research into novel uses of computer technology; the World Wide Web Consortium standards organization was founded at The Laboratory for Computer Science in 1994 by Tim Berners-Lee; The MIT OpenCourseWare project has made course materials for over 2,000 Massachusetts Institute of Technology classes available online free of charge since 2002; and The One Laptop per Child initiative to expand computer education and connectivity to children worldwide was launched in 2005.

    The Massachusetts Institute of Technology was named a sea-grant college in 1976 to support its programs in oceanography and marine sciences and was named a space-grant college in 1989 to support its aeronautics and astronautics programs. Despite diminishing government financial support over the past quarter century, MIT launched several successful development campaigns to significantly expand the campus: new dormitories and athletics buildings on west campus; The Tang Center for Management Education; several buildings in the northeast corner of campus supporting research into biology, brain and cognitive sciences, genomics, biotechnology, and cancer research; and a number of new “backlot” buildings on Vassar Street including The Stata Center. Construction on campus in the 2000s included expansions of the Media Lab, the Sloan School’s eastern campus, and graduate residences in the northwest. In 2006, President Hockfield launched The MIT Energy Research Council to investigate the interdisciplinary challenges posed by increasing global energy consumption.

    In 2001, inspired by the open source and open access movements, The Massachusetts Institute of Technology launched OpenCourseWare to make the lecture notes, problem sets, syllabi, exams, and lectures from the great majority of its courses available online for no charge, though without any formal accreditation for coursework completed. While the cost of supporting and hosting the project is high, OCW expanded in 2005 to include other universities as a part of the OpenCourseWare Consortium, which currently includes more than 250 academic institutions with content available in at least six languages. In 2011, The Massachusetts Institute of Technology announced it would offer formal certification (but not credits or degrees) to online participants completing coursework in its “MITx” program, for a modest fee. The “edX” online platform supporting MITx was initially developed in partnership with Harvard and its analogous “Harvardx” initiative. The courseware platform is open source, and other universities have already joined and added their own course content. In March 2009 The Massachusetts Institute of Technology faculty adopted an open-access policy to make its scholarship publicly accessible online.

    The Massachusetts Institute of Technology has its own police force. Three days after the Boston Marathon bombing of April 2013, MIT Police patrol officer Sean Collier was fatally shot by the suspects Dzhokhar and Tamerlan Tsarnaev, setting off a violent manhunt that shut down the campus and much of the Boston metropolitan area for a day. One week later, Collier’s memorial service was attended by more than 10,000 people, in a ceremony hosted by The Massachusetts Institute of Technology (US) community with thousands of police officers from the New England region and Canada. On November 25, 2013, The Massachusetts Institute of Technology announced the creation of The Collier Medal, to be awarded annually to “an individual or group that embodies the character and qualities that Officer Collier exhibited as a member of The Massachusetts Institute of Technology community and in all aspects of his life”. The announcement further stated that “Future recipients of the award will include those whose contributions exceed the boundaries of their profession, those who have contributed to building bridges across the community, and those who consistently and selflessly perform acts of kindness”.

    In September 2017, the school announced the creation of an artificial intelligence research lab called The MIT-IBM Watson AI Lab. IBM will spend $240 million over the next decade, and the lab will be staffed by MIT and IBM scientists. In October 2018 MIT announced that it would open a new Schwarzman College of Computing dedicated to the study of artificial intelligence, named after lead donor and The Blackstone Group CEO Stephen Schwarzman. The focus of the new college is to study not just AI, but interdisciplinary AI education, and how AI can be used in fields as diverse as history and biology. The cost of buildings and new faculty for the new college is expected to be $1 billion upon completion.

    The Caltech/MIT Advanced aLIGO (US) was designed and constructed by a team of scientists from California Institute of Technology (US), Massachusetts Institute of Technology, and industrial contractors, and funded by The National Science Foundation (US).

    It was designed to open the field of gravitational-wave astronomy through the detection of gravitational waves predicted by general relativity. Gravitational waves were detected for the first time by the LIGO detector in 2015. For contributions to the LIGO detector and the observation of gravitational waves, two Caltech physicists, Kip Thorne and Barry Barish, and The Massachusetts Institute of Technology (US) physicist Rainer Weiss won the Nobel Prize in physics in 2017. Weiss, who is also a Massachusetts Institute of Technology graduate, designed the laser interferometric technique, which served as the essential blueprint for the LIGO.

    The mission of The Massachusetts Institute of Technology is to advance knowledge and educate students in science, technology, and other areas of scholarship that will best serve the nation and the world in the twenty-first century. We seek to develop in each member of the Massachusetts Institute of Technology community the ability and passion to work wisely, creatively, and effectively for the betterment of humankind.

     
  • richardmitnick 12:31 pm on August 27, 2021 Permalink | Reply
    Tags: "SURP Student Spotlight-Sina Babaei Zadeh", , Mathematics,   

    From Dunlap Institute for Astronomy and Astrophysics (CA) : “SURP Student Spotlight-Sina Babaei Zadeh” 

    From Dunlap Institute for Astronomy and Astrophysics (CA)

    At

    University of Toronto (CA)

    8.24.21

    1
    Originally from Toronto, Sina is going into his second year of math and astrophysics double major at Western University (CA) this fall. In addition to his SURP grant, he was awarded the Aurora Borealis Fellowship.

    What made you decide to participate in SURP?

    I have always wanted to participate in scientific research. I was fortunate enough to be offered several amazing research opportunities this summer (e.g. CGCS and U of T SURF). I chose SURP because of its attractive resources, especially its mentors. Moreover, SURP offered a variety of events in addition to research such as weekly astronomy seminars and even a free summer school in coding. Truly, I chose SURP because it is not an ordinary undergraduate research project, but a summer in an astronomer’s shoes.

    What is your favourite thing about SURP?

    This is a hard question as there are a lot of top things, but if I had to select one, I would choose the diversity at every level from mentors to administrators to SURP students themselves. There are people from all around the world with unique backgrounds and views. This allows you to experience and learn about research topics and methods in North America, but also from other continents. For instance, I got to work with a UK based simulation- one of the largest in the world- because one of my mentors had completed his education in the UK and thus had familiarity with it. This diversity is also great for learning about other cultures and perspectives, something that I highly value.

    Can you tell us about your research project?

    In our project, we have utilized the EAGLE database, a very large simulation, containing data on about a million galaxies and their relative properties such as SFR (star formation rate) to test if we can make predictions about the properties of a particular galaxy based on its assembly history. Specifically, we grouped galaxies based on the closeness of SFH (star formation history) to exponential functions governed by a single parameter. This allowed us to draw conclusions and trends about the differences of each group when compared to each other. Both our approaches will enable us to predict how galaxies like our own Milky Way may look like in future and predict their early history.

    Can you explain how SURP has perhaps been different from your undergrad work?

    One obvious difference is the learning environment. In regular classes, the student to teacher ratio is very high, but in SURP it can be even smaller than one (in my case it was 1:3)! This allows you to work with top scientists very closely and actually feel part of the researcher community. Furthermore, the environment is much more relaxed and you can just chat with your mentors anytime that you want whether it’s a research question, or just social chatter! Although research is open ended and can be much more intimidating than reading undergraduate textbooks, the support is also at a higher level and as long as you are motivated, it will be much more valuable than normal courses. Simply, the maximum you can do in an undergraduate class is to achieve a perfect grade, but at SURP the maximum is virtually limitless.

    What are your plans for the future?

    I want to keep my options open. After SURP, graduate school in astronomy is a likely possibility. All I know is that I feel tremendously grateful to have had the opportunity to participate in SURP and attend university in general. I really want to try my best to be in the position of helping the next generation in a few years, especially my fellow black Canadians. This will allow me to give back some of the things that I have used, and enable many other students to have quality experiences like I did at SURP. To put it simply, SURP has not given me a glimpse of what a career in astronomy will look like, it has also expanded my vision and made me a more mature individual.

    See the full article here .


    five-ways-keep-your-child-safe-school-shootings

    Please help promote STEM in your local schools.


    Stem Education Coalition

    Dunlap Institute campus

    The Dunlap Institute for Astronomy & Astrophysics (CA) at University of Toronto (CA) is an endowed research institute with nearly 70 faculty, postdocs, students and staff, dedicated to innovative technology, ground-breaking research, world-class training, and public engagement. The research themes of its faculty and Dunlap Fellows span the Universe and include: optical, infrared and radio instrumentation; Dark Energy; large-scale structure; the Cosmic Microwave Background; the interstellar medium; galaxy evolution; cosmic magnetism; and time-domain science.

    The Dunlap Institute (CA), University of Toronto Department of Astronomy & Astrophysics (CA), Canadian Institute for Theoretical Astrophysics (CA), and Centre for Planetary Sciences (CA) comprise the leading centre for astronomical research in Canada, at the leading research university in the country, the University of Toronto (CA).

    The Dunlap Institute (CA) is committed to making its science, training and public outreach activities productive and enjoyable for everyone, regardless of gender, sexual orientation, disability, physical appearance, body size, race, nationality or religion.

    Our work is greatly enhanced through collaborations with the Department of Astronomy & Astrophysics (CA), Canadian Institute for Theoretical Astrophysics (CA), David Dunlap Observatory (CA), Ontario Science Centre (CA), Royal Astronomical Society of Canada (CA), the Toronto Public Library (CA), and many other partners.

    NIROSETI team from left to right Rem Stone UCO Lick Observatory Dan Werthimer, UC Berkeley; Jérôme Maire, U Toronto; Shelley Wright, UCSD; Patrick Dorval, U Toronto; Richard Treffers, Starman Systems. (Image by Laurie Hatch).

    The University of Toronto(CA) is a public research university in Toronto, Ontario, Canada, located on the grounds that surround Queen’s Park. It was founded by royal charter in 1827 as King’s College, the oldest university in the province of Ontario.

    Originally controlled by the Church of England, the university assumed its present name in 1850 upon becoming a secular institution.

    As a collegiate university, it comprises eleven colleges each with substantial autonomy on financial and institutional affairs and significant differences in character and history. The university also operates two satellite campuses located in Scarborough and Mississauga.

    University of Toronto has evolved into Canada’s leading institution of learning, discovery and knowledge creation. We are proud to be one of the world’s top research-intensive universities, driven to invent and innovate.

    Our students have the opportunity to learn from and work with preeminent thought leaders through our multidisciplinary network of teaching and research faculty, alumni and partners.

    The ideas, innovations and actions of more than 560,000 graduates continue to have a positive impact on the world.

    Academically, the University of Toronto is noted for movements and curricula in literary criticism and communication theory, known collectively as the Toronto School.

    The university was the birthplace of insulin and stem cell research, and was the site of the first electron microscope in North America; the identification of the first black hole Cygnus X-1; multi-touch technology, and the development of the theory of NP-completeness.

    The university was one of several universities involved in early research of deep learning. It receives the most annual scientific research funding of any Canadian university and is one of two members of the Association of American Universities (US) outside the United States, the other being McGill(CA).

    The Varsity Blues are the athletic teams that represent the university in intercollegiate league matches, with ties to gridiron football, rowing and ice hockey. The earliest recorded instance of gridiron football occurred at University of Toronto’s University College in November 1861.

    The university’s Hart House is an early example of the North American student centre, simultaneously serving cultural, intellectual, and recreational interests within its large Gothic-revival complex.

    The University of Toronto has educated three Governors General of Canada, four Prime Ministers of Canada, three foreign leaders, and fourteen Justices of the Supreme Court. As of March 2019, ten Nobel laureates, five Turing Award winners, 94 Rhodes Scholars, and one Fields Medalist have been affiliated with the university.

    Early history

    The founding of a colonial college had long been the desire of John Graves Simcoe, the first Lieutenant-Governor of Upper Canada and founder of York, the colonial capital. As an University of Oxford (UK)-educated military commander who had fought in the American Revolutionary War, Simcoe believed a college was needed to counter the spread of republicanism from the United States. The Upper Canada Executive Committee recommended in 1798 that a college be established in York.

    On March 15, 1827, a royal charter was formally issued by King George IV, proclaiming “from this time one College, with the style and privileges of a University … for the education of youth in the principles of the Christian Religion, and for their instruction in the various branches of Science and Literature … to continue for ever, to be called King’s College.” The granting of the charter was largely the result of intense lobbying by John Strachan, the influential Anglican Bishop of Toronto who took office as the college’s first president. The original three-storey Greek Revival school building was built on the present site of Queen’s Park.

    Under Strachan’s stewardship, King’s College was a religious institution closely aligned with the Church of England and the British colonial elite, known as the Family Compact. Reformist politicians opposed the clergy’s control over colonial institutions and fought to have the college secularized. In 1849, after a lengthy and heated debate, the newly elected responsible government of the Province of Canada voted to rename King’s College as the University of Toronto and severed the school’s ties with the church. Having anticipated this decision, the enraged Strachan had resigned a year earlier to open Trinity College as a private Anglican seminary. University College was created as the nondenominational teaching branch of the University of Toronto. During the American Civil War the threat of Union blockade on British North America prompted the creation of the University Rifle Corps which saw battle in resisting the Fenian raids on the Niagara border in 1866. The Corps was part of the Reserve Militia lead by Professor Henry Croft.

    Established in 1878, the School of Practical Science was the precursor to the Faculty of Applied Science and Engineering which has been nicknamed Skule since its earliest days. While the Faculty of Medicine opened in 1843 medical teaching was conducted by proprietary schools from 1853 until 1887 when the faculty absorbed the Toronto School of Medicine. Meanwhile the university continued to set examinations and confer medical degrees. The university opened the Faculty of Law in 1887, followed by the Faculty of Dentistry in 1888 when the Royal College of Dental Surgeons became an affiliate. Women were first admitted to the university in 1884.

    A devastating fire in 1890 gutted the interior of University College and destroyed 33,000 volumes from the library but the university restored the building and replenished its library within two years. Over the next two decades a collegiate system took shape as the university arranged federation with several ecclesiastical colleges including Strachan’s Trinity College in 1904. The university operated the Royal Conservatory of Music from 1896 to 1991 and the Royal Ontario Museum from 1912 to 1968; both still retain close ties with the university as independent institutions. The University of Toronto Press was founded in 1901 as Canada’s first academic publishing house. The Faculty of Forestry founded in 1907 with Bernhard Fernow as dean was Canada’s first university faculty devoted to forest science. In 1910, the Faculty of Education opened its laboratory school, the University of Toronto Schools.

    World wars and post-war years

    The First and Second World Wars curtailed some university activities as undergraduate and graduate men eagerly enlisted. Intercollegiate athletic competitions and the Hart House Debates were suspended although exhibition and interfaculty games were still held. The David Dunlap Observatory in Richmond Hill opened in 1935 followed by the University of Toronto Institute for Aerospace Studies in 1949. The university opened satellite campuses in Scarborough in 1964 and in Mississauga in 1967. The university’s former affiliated schools at the Ontario Agricultural College and Glendon Hall became fully independent of the University of Toronto and became part of University of Guelph (CA) in 1964 and York University (CA) in 1965 respectively. Beginning in the 1980s reductions in government funding prompted more rigorous fundraising efforts.

    Since 2000

    In 2000 Kin-Yip Chun was reinstated as a professor of the university after he launched an unsuccessful lawsuit against the university alleging racial discrimination. In 2017 a human rights application was filed against the University by one of its students for allegedly delaying the investigation of sexual assault and being dismissive of their concerns. In 2018 the university cleared one of its professors of allegations of discrimination and antisemitism in an internal investigation after a complaint was filed by one of its students.

    The University of Toronto was the first Canadian university to amass a financial endowment greater than c. $1 billion in 2007. On September 24, 2020 the university announced a $250 million gift to the Faculty of Medicine from businessman and philanthropist James C. Temerty- the largest single philanthropic donation in Canadian history. This broke the previous record for the school set in 2019 when Gerry Schwartz and Heather Reisman jointly donated $100 million for the creation of a 750,000-square foot innovation and artificial intelligence centre.

    Research

    Since 1926 the University of Toronto has been a member of the Association of American Universities (US) a consortium of the leading North American research universities. The university manages by far the largest annual research budget of any university in Canada with sponsored direct-cost expenditures of $878 million in 2010. In 2018 the University of Toronto was named the top research university in Canada by Research Infosource with a sponsored research income (external sources of funding) of $1,147.584 million in 2017. In the same year the university’s faculty averaged a sponsored research income of $428,200 while graduate students averaged a sponsored research income of $63,700. The federal government was the largest source of funding with grants from the Canadian Institutes of Health Research; the Natural Sciences and Engineering Research Council; and the Social Sciences and Humanities Research Council amounting to about one-third of the research budget. About eight percent of research funding came from corporations- mostly in the healthcare industry.

    The first practical electron microscope was built by the physics department in 1938. During World War II the university developed the G-suit- a life-saving garment worn by Allied fighter plane pilots later adopted for use by astronauts.Development of the infrared chemiluminescence technique improved analyses of energy behaviours in chemical reactions. In 1963 the asteroid 2104 Toronto was discovered in the David Dunlap Observatory (CA) in Richmond Hill and is named after the university. In 1972 studies on Cygnus X-1 led to the publication of the first observational evidence proving the existence of black holes. Toronto astronomers have also discovered the Uranian moons of Caliban and Sycorax; the dwarf galaxies of Andromeda I, II and III; and the supernova SN 1987A. A pioneer in computing technology the university designed and built UTEC- one of the world’s first operational computers- and later purchased Ferut- the second commercial computer after UNIVAC I. Multi-touch technology was developed at Toronto with applications ranging from handheld devices to collaboration walls. The AeroVelo Atlas which won the Igor I. Sikorsky Human Powered Helicopter Competition in 2013 was developed by the university’s team of students and graduates and was tested in Vaughan.

    The discovery of insulin at the University of Toronto in 1921 is considered among the most significant events in the history of medicine. The stem cell was discovered at the university in 1963 forming the basis for bone marrow transplantation and all subsequent research on adult and embryonic stem cells. This was the first of many findings at Toronto relating to stem cells including the identification of pancreatic and retinal stem cells. The cancer stem cell was first identified in 1997 by Toronto researchers who have since found stem cell associations in leukemia; brain tumors; and colorectal cancer. Medical inventions developed at Toronto include the glycaemic index; the infant cereal Pablum; the use of protective hypothermia in open heart surgery; and the first artificial cardiac pacemaker. The first successful single-lung transplant was performed at Toronto in 1981 followed by the first nerve transplant in 1988; and the first double-lung transplant in 1989. Researchers identified the maturation promoting factor that regulates cell division and discovered the T-cell receptor which triggers responses of the immune system. The university is credited with isolating the genes that cause Fanconi anemia; cystic fibrosis; and early-onset Alzheimer’s disease among numerous other diseases. Between 1914 and 1972 the university operated the Connaught Medical Research Laboratories- now part of the pharmaceutical corporation Sanofi-Aventis. Among the research conducted at the laboratory was the development of gel electrophoresis.

    The University of Toronto is the primary research presence that supports one of the world’s largest concentrations of biotechnology firms. More than 5,000 principal investigators reside within 2 kilometres (1.2 mi) from the university grounds in Toronto’s Discovery District conducting $1 billion of medical research annually. MaRS Discovery District is a research park that serves commercial enterprises and the university’s technology transfer ventures. In 2008, the university disclosed 159 inventions and had 114 active start-up companies. Its SciNet Consortium operates the most powerful supercomputer in Canada.

     
  • richardmitnick 3:00 pm on August 25, 2021 Permalink | Reply
    Tags: "CAMERA Mathematicians Build an Algorithm to ‘Do the Twist’", A mathematical algorithm to decipher the rotational dynamics of twisting particles in large complex systems., , , Mathematics, Studying the properties of suspensions and solutions of colloids; macromolecules; and polymers., XPCS works by focusing a coherent beam of X-rays to capture light scattered off of particles in suspension.,   

    From DOE’s Lawrence Berkeley National Laboratory (US): “CAMERA Mathematicians Build an Algorithm to ‘Do the Twist’” 

    From DOE’s Lawrence Berkeley National Laboratory (US)

    August 18, 2021

    New Approach Extracts Rotational Diffusion from X-ray Photon Correlation Spectroscopy Experiments.

    Mathematicians at the Center for Advanced Mathematics for Energy Research Applications (CAMERA) at Lawrence Berkeley National Laboratory (Berkeley Lab) have developed a mathematical algorithm to decipher the rotational dynamics of twisting particles in large complex systems from the X-ray scattering patterns observed in highly sophisticated X-ray photon correlation spectroscopy (XPCS) experiments.

    1
    Schematic illustration of the XPCS experiments. The translation and rotation of the particles within the scattering volume leads to variation of the speckle patterns shown on the right. (While the grainy, noise-like texture makes these images appear visually similar, the MTECS algorithm is able to detect and analyze tiny variations between patterns.)

    These experiments — designed to study the properties of suspensions and solutions of colloids, macromolecules and polymers — have been established as key scientific drivers to many of the ongoing coherent light source upgrades occurring within the U.S. Department of Energy (DOE). The new mathematical methods, developed by the CAMERA team of Zixi Hu, Jeffrey Donatelli, and James Sethian, have the potential to reveal far more information about the function and properties of complex materials than was previously possible.

    Particles in a suspension undergo Brownian motion, jiggling around as they move (translate) and spin (rotate). The sizes of these random fluctuations depend on the shape and structure of the materials and contain information about dynamics, with applications across molecular biology, drug discovery, and materials science.

    XPCS works by focusing a coherent beam of X-rays to capture light scattered off of particles in suspension. A detector picks up the resulting speckle patterns, which contain several tiny fluctuations in the signal that encode detailed information about the dynamics of the observed system. To capitalize on this capability, the upcoming coherent light source upgrades at Berkeley Lab’s Advanced Light Source (ALS), Argonne’s Advanced Photon Source (APS), and SLAC’s Linac Coherent Light Source are all planning some of the world’s most advanced XPCS experiments, taking advantage of the unprecedented coherence and brightness.

    But once you collect the data from all these images, how do you get any useful information out of them? A workhorse technique to extract dynamical information from XPCS is to compute what’s known as the temporal autocorrelation, which measures how the pixels in the speckle patterns change after a certain passage of time. The autocorrelation function stitches the still images together, just as an old-time movie comes to life as closely related postcard images fly by.

    Current algorithms have mainly been limited to extracting translational motions; think of a Pogo stick jumping from spot to spot. However, no previous algorithms were capable of extracting “rotational diffusion” information about how structures spin and rotate — information that is critical to understanding the function and dynamical properties of a physical system. Getting to this hidden information is a major challenge.
    Twisting the Light Away

    A breakthrough came when experts came together for a CAMERA workshop on XPCS in February 2019 to discuss critical emerging needs in the field. Extracting rotational diffusion was a key goal, and Hu, a UC Berkeley math graduate student; Donatelli, the CAMERA Lead for Mathematics; and Sethian, Professor of Mathematics at UC Berkeley and CAMERA Director, teamed up to tackle the problem head on.

    The result of their work is a powerful new mathematical and algorithmic approach to extract rotational information, now working in 2D and easily scalable to 3D. With remarkably few images (less than 4,000), the method can easily predict simulated rotational diffusion coefficients to within a few percent. Details of the algorithm were published August 18 in the PNAS.

    The key idea is to go beyond the standard autocorrelation function, instead seeking the extra information about rotation contained in angular-temporal cross-correlation functions, which compare how pixels change in both time and space. This is a major jump in mathematical complexity: simple data matrices turn into 4-way data tensors, and the theory relating the rotational information to these tensors involves advanced harmonic analysis, linear algebra, and tensor analysis. To relate the desired rotational information to the data, Hu developed a highly sophisticated mathematical model that describes how the angular-temporal correlations behave as a function of the rotational dynamics from this new complex set of equations.

    “There were lots of layered mysteries to unravel in order to build a good mathematical and algorithmic framework to solve the problem,” said Hu. “There was information related to both static structures and to dynamic properties, and these properties needed to be systematically exploited to build a consistent framework. Taken together, they present a wonderful opportunity to weave together many mathematical ideas. Getting this approach to pick up useful information out of what seems at first glance to be awfully noisy was great fun.”

    However, solving this set of equations to recover the rotational dynamics is challenging, as it consists of several layers of different types of mathematical problems that are difficult to solve all at once. To tackle this challenge, the team built on Donatelli’s earlier work on Multi-Tiered Iterative Projections (M-TIP), which is designed to solve complex inverse problems where the goal is to find the input that produces an observed output. The idea of M-TIP is to break a complex problem into subparts, using the best inversion/pseudoinversion you can for each subpart, and iterate through those subsolutions until they converge to a solution that solves all parts of the problem.

    Hu and his colleagues took these ideas and built a sister method, “Multi-Tiered Estimation for Correlation Spectroscopy (M-TECS),” solving the complex layered set of equations through systematic substeps.

    2
    Schematic illustration of the M-TECS algorithm for determining the rotational diffusion coefficient of a dynamical system from its XPCS cross-correlation data. M-TECS works by splitting up the inversion into subparts, each of which of efficient mathematical tricks for inverting/pseudoinverting, and then iterating over those subparts until convergence to the correct solution.

    “The powerful thing about the M-TECS approach is that it exploits the fact that the problem can be separated into high-dimensional linear parts and low-dimensional nonlinear and nonconvex parts, each of which have efficient solutions on their own, but they would turn into an exceedingly difficult optimization problem if they were instead to be solved for all at once,” said Donatelli.

    “This is what enables M-TECS to efficiently determine rotational dynamics from such a complex system of equations, whereas standard optimization approaches would run into trouble both in terms of convergence and computational cost.”

    Opening the Door to New Experiments

    “XPCS is a powerful technique that will feature prominently in the ALS upgrade. This work opens up a new dimension to XPCS, and will allow us to explore the dynamics of complex materials such as rotating molecules inside water channels,” said Alexander Hexemer, Program Lead for Computing at the ALS.

    Hu, who won UC Berkeley’s Bernard Friedman Prize for this work, has joined CAMERA — part of Berkeley Lab’s Computational Research Division — as its newest member. “This sort of mathematical and algorithmic co-design is the hallmark of good applied mathematics, in which new mathematics plays a pivotal role in solving practical problems at the forefront of scientific inquiry,” said Sethian.

    The CAMERA team is currently working with beamline scientists at the ALS and APS to design new XPCS experiments that can fully leverage the team’s mathematical and algorithmic approach to study new rotational dynamics properties from important materials. The team is also working on extending their mathematical and algorithmic framework work to recover more general types of dynamical properties from XPCS, as well as apply these methods to other correlation imaging technologies.

    This work is supported by CAMERA, which is jointly funded by the DOE Office of Science Advanced Scientific Computing Research (US) and the Office of Basic Energy Sciences, both within the Department of Energy’s (US) Office of Science.

    See the full article here .

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    Please help promote STEM in your local schools.

    Stem Education Coalition


    Bringing Science Solutions to the World

    In the world of science, Lawrence Berkeley National Laboratory (Berkeley Lab) (US) is synonymous with “excellence.” Thirteen Nobel prizes are associated with Berkeley Lab. Seventy Lab scientists are members of the National Academy of Sciences (NAS), one of the highest honors for a scientist in the United States. Thirteen of our scientists have won the National Medal of Science, our nation’s highest award for lifetime achievement in fields of scientific research. Eighteen of our engineers have been elected to the National Academy of Engineering, and three of our scientists have been elected into the Institute of Medicine. In addition, Berkeley Lab has trained thousands of university science and engineering students who are advancing technological innovations across the nation and around the world.

    Berkeley Lab is a member of the national laboratory system supported by the U.S. Department of Energy through its Office of Science. It is managed by the University of California (US) and is charged with conducting unclassified research across a wide range of scientific disciplines. Located on a 202-acre site in the hills above the UC Berkeley campus that offers spectacular views of the San Francisco Bay, Berkeley Lab employs approximately 3,232 scientists, engineers and support staff. The Lab’s total costs for FY 2014 were $785 million. A recent study estimates the Laboratory’s overall economic impact through direct, indirect and induced spending on the nine counties that make up the San Francisco Bay Area to be nearly $700 million annually. The Lab was also responsible for creating 5,600 jobs locally and 12,000 nationally. The overall economic impact on the national economy is estimated at $1.6 billion a year. Technologies developed at Berkeley Lab have generated billions of dollars in revenues, and thousands of jobs. Savings as a result of Berkeley Lab developments in lighting and windows, and other energy-efficient technologies, have also been in the billions of dollars.

    Berkeley Lab was founded in 1931 by Ernest Orlando Lawrence, a University of California-Berkeley (US) physicist who won the 1939 Nobel Prize in physics for his invention of the cyclotron, a circular particle accelerator that opened the door to high-energy physics. It was Lawrence’s belief that scientific research is best done through teams of individuals with different fields of expertise, working together. His teamwork concept is a Berkeley Lab legacy that continues today.

    History

    1931–1941

    The laboratory was founded on August 26, 1931, by Ernest Lawrence, as the Radiation Laboratory of the University of California, Berkeley, associated with the Physics Department. It centered physics research around his new instrument, the cyclotron, a type of particle accelerator for which he was awarded the Nobel Prize in Physics in 1939.

    LBNL 88 inch cyclotron.


    Throughout the 1930s, Lawrence pushed to create larger and larger machines for physics research, courting private philanthropists for funding. He was the first to develop a large team to build big projects to make discoveries in basic research. Eventually these machines grew too large to be held on the university grounds, and in 1940 the lab moved to its current site atop the hill above campus. Part of the team put together during this period includes two other young scientists who went on to establish large laboratories; J. Robert Oppenheimer founded DOE’s Los Alamos Laboratory (US), and Robert Wilson founded Fermi National Accelerator Laboratory(US).

    1942–1950

    Leslie Groves visited Lawrence’s Radiation Laboratory in late 1942 as he was organizing the Manhattan Project, meeting J. Robert Oppenheimer for the first time. Oppenheimer was tasked with organizing the nuclear bomb development effort and founded today’s Los Alamos National Laboratory to help keep the work secret. At the RadLab, Lawrence and his colleagues developed the technique of electromagnetic enrichment of uranium using their experience with cyclotrons. The “calutrons” (named after the University) became the basic unit of the massive Y-12 facility in Oak Ridge, Tennessee. Lawrence’s lab helped contribute to what have been judged to be the three most valuable technology developments of the war (the atomic bomb, proximity fuse, and radar). The cyclotron, whose construction was stalled during the war, was finished in November 1946. The Manhattan Project shut down two months later.

    1951–2018

    After the war, the Radiation Laboratory became one of the first laboratories to be incorporated into the Atomic Energy Commission (AEC) (now Department of Energy (US). The most highly classified work remained at Los Alamos, but the RadLab remained involved. Edward Teller suggested setting up a second lab similar to Los Alamos to compete with their designs. This led to the creation of an offshoot of the RadLab (now the Lawrence Livermore National Laboratory (US)) in 1952. Some of the RadLab’s work was transferred to the new lab, but some classified research continued at Berkeley Lab until the 1970s, when it became a laboratory dedicated only to unclassified scientific research.

    Shortly after the death of Lawrence in August 1958, the UC Radiation Laboratory (both branches) was renamed the Lawrence Radiation Laboratory. The Berkeley location became the Lawrence Berkeley Laboratory in 1971, although many continued to call it the RadLab. Gradually, another shortened form came into common usage, LBNL. Its formal name was amended to Ernest Orlando Lawrence Berkeley National Laboratory in 1995, when “National” was added to the names of all DOE labs. “Ernest Orlando” was later dropped to shorten the name. Today, the lab is commonly referred to as “Berkeley Lab”.

    The Alvarez Physics Memos are a set of informal working papers of the large group of physicists, engineers, computer programmers, and technicians led by Luis W. Alvarez from the early 1950s until his death in 1988. Over 1700 memos are available on-line, hosted by the Laboratory.

    The lab remains owned by the Department of Energy (US), with management from the University of California (US). Companies such as Intel were funding the lab’s research into computing chips.

    Science mission

    From the 1950s through the present, Berkeley Lab has maintained its status as a major international center for physics research, and has also diversified its research program into almost every realm of scientific investigation. Its mission is to solve the most pressing and profound scientific problems facing humanity, conduct basic research for a secure energy future, understand living systems to improve the environment, health, and energy supply, understand matter and energy in the universe, build and safely operate leading scientific facilities for the nation, and train the next generation of scientists and engineers.

    The Laboratory’s 20 scientific divisions are organized within six areas of research: Computing Sciences; Physical Sciences; Earth and Environmental Sciences; Biosciences; Energy Sciences; and Energy Technologies. Berkeley Lab has six main science thrusts: advancing integrated fundamental energy science; integrative biological and environmental system science; advanced computing for science impact; discovering the fundamental properties of matter and energy; accelerators for the future; and developing energy technology innovations for a sustainable future. It was Lawrence’s belief that scientific research is best done through teams of individuals with different fields of expertise, working together. His teamwork concept is a Berkeley Lab tradition that continues today.

    Berkeley Lab operates five major National User Facilities for the DOE Office of Science (US):

    The Advanced Light Source (ALS) is a synchrotron light source with 41 beam lines providing ultraviolet, soft x-ray, and hard x-ray light to scientific experiments.

    LBNL/ALS


    The ALS is one of the world’s brightest sources of soft x-rays, which are used to characterize the electronic structure of matter and to reveal microscopic structures with elemental and chemical specificity. About 2,500 scientist-users carry out research at ALS every year. Berkeley Lab is proposing an upgrade of ALS which would increase the coherent flux of soft x-rays by two-three orders of magnitude.

    The DOE Joint Genome Institute (US) supports genomic research in support of the DOE missions in alternative energy, global carbon cycling, and environmental management. The JGI’s partner laboratories are Berkeley Lab, DOE’s Lawrence Livermore National Laboratory (US), DOE’s Oak Ridge National Laboratory (US)(ORNL), DOE’s Pacific Northwest National Laboratory (US) (PNNL), and the HudsonAlpha Institute for Biotechnology (US). The JGI’s central role is the development of a diversity of large-scale experimental and computational capabilities to link sequence to biological insights relevant to energy and environmental research. Approximately 1,200 scientist-users take advantage of JGI’s capabilities for their research every year.

    The LBNL Molecular Foundry (US) [above] is a multidisciplinary nanoscience research facility. Its seven research facilities focus on Imaging and Manipulation of Nanostructures; Nanofabrication; Theory of Nanostructured Materials; Inorganic Nanostructures; Biological Nanostructures; Organic and Macromolecular Synthesis; and Electron Microscopy. Approximately 700 scientist-users make use of these facilities in their research every year.

    The DOE’s NERSC National Energy Research Scientific Computing Center (US) is the scientific computing facility that provides large-scale computing for the DOE’s unclassified research programs. Its current systems provide over 3 billion computational hours annually. NERSC supports 6,000 scientific users from universities, national laboratories, and industry.

    DOE’s NERSC National Energy Research Scientific Computing Center(US) at Lawrence Berkeley National Laboratory

    The Genepool system is a cluster dedicated to the DOE Joint Genome Institute’s computing needs. Denovo is a smaller test system for Genepool that is primarily used by NERSC staff to test new system configurations and software.

    PDSF is a networked distributed computing cluster designed primarily to meet the detector simulation and data analysis requirements of physics, astrophysics and nuclear science collaborations.

    NERSC is a DOE Office of Science User Facility.

    The DOE’s Energy Science Network (US) is a high-speed network infrastructure optimized for very large scientific data flows. ESNet provides connectivity for all major DOE sites and facilities, and the network transports roughly 35 petabytes of traffic each month.

    Berkeley Lab is the lead partner in the DOE’s Joint Bioenergy Institute (US) (JBEI), located in Emeryville, California. Other partners are the DOE’s Sandia National Laboratory (US), the University of California (UC) campuses of Berkeley and Davis, the Carnegie Institution for Science (US), and DOE’s Lawrence Livermore National Laboratory (US) (LLNL). JBEI’s primary scientific mission is to advance the development of the next generation of biofuels – liquid fuels derived from the solar energy stored in plant biomass. JBEI is one of three new U.S. Department of Energy (DOE) Bioenergy Research Centers (BRCs).

    Berkeley Lab has a major role in two DOE Energy Innovation Hubs. The mission of the Joint Center for Artificial Photosynthesis (JCAP) is to find a cost-effective method to produce fuels using only sunlight, water, and carbon dioxide. The lead institution for JCAP is the California Institute of Technology (US) and Berkeley Lab is the second institutional center. The mission of the Joint Center for Energy Storage Research (JCESR) is to create next-generation battery technologies that will transform transportation and the electricity grid. DOE’s Argonne National Laboratory (US) leads JCESR and Berkeley Lab is a major partner.

     
  • richardmitnick 12:25 pm on August 20, 2021 Permalink | Reply
    Tags: "How Big Data Carried Graph Theory Into New Dimensions", , Hypergraph, Markov chain, Mathematics, New kinds of network models that can find complex structures and signals in the noise of big data., , Simplicial complexes, Tensors,   

    From Quanta Magazine (US) : “How Big Data Carried Graph Theory Into New Dimensions” 

    From Quanta Magazine (US)

    August 19, 2021
    Stephen Ornes

    1
    Mike Hughes for Quanta Magazine.

    The mathematical language for talking about connections, which usually depends on networks — vertices (dots) and edges (lines connecting them) — has been an invaluable way to model real-world phenomena since at least the 18th century. But a few decades ago, the emergence of giant data sets forced researchers to expand their toolboxes and, at the same time, gave them sprawling sandboxes in which to apply new mathematical insights. Since then, said Josh Grochow, a computer scientist at the University of Colorado-Boulder (US), there’s been an exciting period of rapid growth as researchers have developed new kinds of network models that can find complex structures and signals in the noise of big data.

    Grochow is among a growing chorus of researchers who point out that when it comes to finding connections in big data, graph theory has its limits. A graph represents every relationship as a dyad, or pairwise interaction. However, many complex systems can’t be represented by binary connections alone. Recent progress in the field shows how to move forward.

    Consider trying to forge a network model of parenting. Clearly, each parent has a connection to a child, but the parenting relationship isn’t just the sum of the two links, as graph theory might model it. The same goes for trying to model a phenomenon like peer pressure.

    “There are many intuitive models. The peer pressure effect on social dynamics is only captured if you already have groups in your data,” said Leonie Neuhäuser of RWTH AACHEN UNIVERSITY [Rheinisch-Westfaelische Technische Hochschule (DE). But binary networks don’t capture group influences.

    Mathematicians and computer scientists use the term “higher-order interactions” to describe these complex ways that group dynamics, rather than binary links, can influence individual behaviors. These mathematical phenomena appear in everything from entanglement interactions in quantum mechanics to the trajectory of a disease spreading through a population. If a pharmacologist wanted to model drug interactions [npj Systems Biology and Applications], for example, graph theory might show how two drugs respond to each other — but what about three? Or four?

    While the tools for exploring these interactions are not new, it’s only in recent years that high-dimensional data sets have become an engine for discovery, giving mathematicians and network theorists new ideas. These efforts have yielded interesting results about the limits of graphs and the possibilities of scaling up.

    “Now we know that the network is just the shadow of the thing,” Grochow said. If a data set has a complex underlying structure, then modeling it as a graph may reveal only a limited projection of the whole story.

    “We’ve realized that the data structures we’ve used to study things, from a mathematical perspective, aren’t quite fitting what we’re seeing in the data,” said the mathematician Emilie Purvine of the DOE’s Pacific Northwest National Laboratory (US).

    Which is why mathematicians, computer scientists and other researchers are increasingly focusing on ways to generalize graph theory — in its many guises — to explore higher-order phenomena. The last few years have brought a torrent of proposed ways to characterize these interactions, and to mathematically verify them in high-dimensional data sets.

    For Purvine, the mathematical exploration of higher-order interactions is like the mapping of new dimensions. “Think about a graph as a foundation on a two-dimensional plot of land,” she said. The three-dimensional buildings that can go on top could vary significantly. “When you’re down at ground level, they look the same, but what you construct on top is different.”

    Enter the Hypergraph

    The search for those higher-dimensional structures is where the math turns especially murky — and interesting. The higher-order analogue of a graph, for example, is called a hypergraph, and instead of edges, it has “hyperedges.” These can connect multiple nodes, which means it can represent multi-way (or multilinear) relationships. Instead of a line, a hyperedge might be seen as a surface, like a tarp staked in three or more places.

    Which is fine, but there’s still a lot we don’t know about how these structures relate to their conventional counterparts. Mathematicians are currently learning which rules of graph theory also apply for higher-order interactions, suggesting new areas of exploration.

    To illustrate the kinds of relationship that a hypergraph can tease out of a big data set — and an ordinary graph can’t — Purvine points to a simple example close to home, the world of scientific publication. Imagine two data sets, each containing papers co-authored by up to three mathematicians; for simplicity, let’s name them A, B and C. One data set contains six papers, with two papers by each of the three distinct pairs (AB, AC and BC). The other contains only two papers total, each co-authored by all three mathematicians (ABC).

    A graph representation of co-authorship, taken from either data set, might look like a triangle, showing that each mathematician (three nodes) had collaborated with the other two (three links). If your only question was who had collaborated with whom, then you wouldn’t need a hypergraph, Purvine said

    But if you did have a hypergraph, you could also answer questions about less obvious structures. A hypergraph of the first set (with six papers), for example, could include hyperedges showing that each mathematician contributed to four papers. A comparison of hypergraphs from the two sets would show that the papers’ authors differed in the first set but was the same in the second.

    Hypergraphs in the Wild

    Such higher-order methods have already proved useful in applied research, such as when ecologists showed how the reintroduction of wolves to Yellowstone National Park in the 1990s triggered changes in biodiversity and in the structure of the food chain. And in one recent paper, Purvine and her colleagues analyzed a database of biological responses to viral infections, using hypergraphs to identify the most critical genes involved. They also showed how those interactions would have been missed by the usual pairwise analysis afforded by graph theory.

    “That’s the kind of power we’re seeing from hypergraphs, to go above and beyond graphs,” said Purvine.

    However, generalizing from graphs to hypergraphs quickly gets complicated. One way to illustrate this is to consider the canonical cut problem from graph theory, which asks: Given two distinct nodes on a graph, what’s the minimum number of edges you can cut to completely sever all connections between the two? Many algorithms can readily find the optimal number of cuts for a given graph.

    But what about cutting a hypergraph? “There are lots of ways of generalizing this notion of a cut to a hypergraph,” said Austin Benson, a mathematician at Cornell University (US). But there’s no one clear solution, he said, because a hyperedge could be severed various ways, creating new groups of nodes.

    Together with two colleagues, Benson recently tried to formalize all the different ways of splitting up a hypergraph. What they found hinted at a variety of computational complexities: For some situations, the problem was readily solved in polynomial time, which basically means a computer could crunch through solutions in a reasonable time. But for others, the problem was basically unsolvable — it was impossible to know for certain whether a solution existed at all.

    “There are still many open questions there,” Benson said. “Some of these impossibility results are interesting because you can’t possibly reduce them to graphs. And on the theory side, if you haven’t reduced it to something you could have found with a graph, it’s showing you that there is something new there.”

    The Mathematical Sandwich

    But the hypergraph isn’t the only way to explore higher-order interactions. Topology — the mathematical study of geometric properties that don’t change when you stretch, compress or otherwise transform objects — offers a more visual approach. When a topologist studies a network, they look for shapes and surfaces and dimensions. They might note that the edge connecting two nodes is one-dimensional and ask about the properties of one-dimensional objects in different networks. Or they might see the two-dimensional triangular surface formed by connecting three nodes and ask similar questions.

    Topologists call these structures simplicial complexes [European Journal of Physics]. These are, effectively, hypergraphs viewed through the framework of topology. Neural networks, which fall into the general category of machine learning, offer a telling example. They’re driven by algorithms designed to mimic how our brains’ neurons process information. Graph neural networks (GNNs), which model connections between things as pairwise connections, excel at inferring data that’s missing from large data sets, but as in other applications, they could miss interactions that only arise from groups of three or more. In recent years, computer scientists have developed simplicial neural networks, which use higher-order complexes to generalize the approach of GNNs to find these effects.

    Simplicial complexes connect topology to graph theory, and, like hypergraphs, they raise compelling mathematical questions that will drive future investigations. For example, in topology, special kinds of subsets of simplicial complexes are also themselves simplicial complexes and therefore have the same properties. If the same held true for a hypergraph, the subsets would include all the hyperedges within — including all the embedded two-way edges.

    But that’s not always the case. “What we’re seeing now is that data falls into this middle ground where not every hyperedge, not every complex interaction, is the same size as every other one,” Purvine said. “You can have a three-way interaction, but not the pairwise interactions.” Big data sets have shown clearly that the group influence often far outstrips the influence of an individual, whether in biological signaling networks or in social behaviors like peer pressure.

    Purvine describes data as filling the middle of a kind of mathematical sandwich, bound on top by these ideas from topology, and underneath by the limitations of graphs. Network theorists are now challenged to find the new rules for higher-order interactions. And for mathematicians, she said, “there’s room to play.”

    Random Walks and Matrices

    That sense of creative “play” extends to other tools as well. There are all sorts of beautiful connections between graphs and other tools for describing data, said Benson. “But as soon as you move to the higher-order setting, these connections are harder to come by.”

    That’s especially clear when you try to consider a higher-dimensional version of a Markov chain, he said. A Markov chain describes a multistage process in which the next stage depends only on an element’s current position; researchers have used Markov models to describe how things like information, energy and even money flow through a system. Perhaps the best-known example of a Markov chain is a random walk, which describes a path where each step is determined randomly from the one before it. A random walk is also a specific graph: Any walk along a graph can be shown as a sequence moving from node to node along links.

    But how to scale up something as simple as a walk? Researchers turn to higher-order Markov chains, which instead of depending only on current position can consider many of the previous states. This approach proved useful for modeling systems like web browsing behavior and airport traffic flows. Benson has ideas for other ways to extend it: He and his colleagues recently described [SIAM Review] a new model for stochastic, or random, processes that combines higher-order Markov chains with another tool called tensors. They tested it against a data set of taxi rides in New York City to see how well it could predict trajectories. The results were mixed: Their model predicted the movement of cabs better than a usual Markov chain, but neither model was very reliable.

    Tensors themselves represent yet another tool for studying higher-order interactions that has come into its own in recent years. To understand tensors, first think of matrices, which organize data into an array of rows and columns. Now imagine matrices made of matrices, or matrices that have not only rows and columns, but also depth or other dimensions of data. These are tensors. If every matrix corresponded to a musical duet, then tensors would include all possible configurations of instruments.

    Tensors are nothing new to physicists, who have long used them to describe, for example, the different possible quantum states of a particle, but network theorists adopted this tool to expand on the power of matrices in high-dimensional data sets. And mathematicians are using them to crack open new classes of problems. Grochow uses tensors to study the isomorphism problem, which essentially asks how you know whether two objects are, in some way, the same. His recent work with Youming Qiao has produced a new way to identify complex problems that might be difficult or impossible to solve.

    How to Hypergraph Responsibly

    Benson’s inconclusive taxi model raises a pervasive question: When do researchers actually need tools like hypergraphs? In many cases, under the right conditions, a hypergraph will deliver the exact same type of predictions and analyses as a graph. “If something is already encapsulated in the network, is it really necessary to model the system [as higher-order]?” asked Michael Schaub of RWTH Aachen University.

    It depends on the data set, he said. “A graph is a good abstraction for a social network, but social networks are so much more. With higher-order systems, there are more ways to model.” Graph theory may show how individuals are connected, for example, but not capture the ways in which clusters of friends on social media influence each other’s behavior.

    The same higher-order interactions won’t emerge in every data set, so new theories are, curiously, driven by the data — which challenges the underlying logical sense that drew Purvine to the field in the first place. “What I love about math is that it’s based in logic and if you follow the right direction, you get to the right answer. But sometimes, when you’re defining whole new areas of math, there’s this subjectivity of what is the right way of doing it,” she says. “And if you don’t recognize that there are multiple ways of doing it, you can maybe drive the community in the wrong direction.”

    Ultimately, Grochow said, these tools represent a kind of freedom, not just allowing researchers to better understand their data, but allowing mathematicians and computer scientists to explore new worlds of possibilities. “There’s endless stuff to explore. It’s interesting and beautiful, and a source of a lot of great questions.”

    See the full article here .


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    Please help promote STEM in your local schools.

    Stem Education Coalition

    Formerly known as Simons Science News, Quanta Magazine (US) is an editorially independent online publication launched by the Simons Foundation to enhance public understanding of science. Why Quanta? Albert Einstein called photons “quanta of light.” Our goal is to “illuminate science.” At Quanta Magazine, scientific accuracy is every bit as important as telling a good story. All of our articles are meticulously researched, reported, edited, copy-edited and fact-checked.

     
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