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  • richardmitnick 8:19 pm on September 7, 2022 Permalink | Reply
    Tags: "Pioneering mathematical formula paves way for exciting advances in the health and energy and food industries", , Mathematics,   

    From The University of Bristol (UK): “Pioneering mathematical formula paves way for exciting advances in the health and energy and food industries” 

    From The University of Bristol (UK)

    9.7.22

    A groundbreaking mathematical equation has been discovered, which could transform medical procedures, natural gas extraction, and plastic packaging production in the future.

    1
    Scientists have discovered a new equation to model exactly diffusive movement through permeable material for the first time. Image credit: University of Bristol.

    The new equation, developed by scientists at the University of Bristol, indicates that diffusive movement through permeable material can be modelled exactly for the very first time. It comes a century after world-leading physicists Albert Einstein and Marian von Smoluchowski derived the first diffusion equation and marks important progress in representing motion for a wide range of entities from microscopic particles and natural organisms to man-made devices.

    Until now, scientists looking at particle motion through porous materials such as biological tissues, polymers, various rocks and sponges, have had to rely on approximations or incomplete perspectives.

    The findings, published today in the journal Physical Review Research [below], provide a novel technique presenting exciting opportunities in a diverse range of settings including health, energy, and the food industry.

    Lead author Toby Kay, who is completing a PhD in Engineering Mathematics, said: “This marks a fundamental step forward since Einstein and Smoluchowski’s studies on diffusion. It revolutionizes the modelling of diffusing entities through complex media of all scales, from cellular components and geological compounds to environmental habitats.

    “Previously, mathematical attempts to represent movement through environments scattered with objects that hinder motion, known as permeable barriers, have been limited. By solving this problem, we are paving the way for exciting advances in many different sectors because permeable barriers are routinely encountered by animals, cellular organisms and humans.”

    Creativity in mathematics takes different forms and one of these is the connection between different levels of description of a phenomenon. In this instance, by representing random motion in a microscopic fashion and then subsequently zooming out to describe the process microscopically, it was possible to find the new equation.

    Further research is needed to apply this mathematical tool to experimental applications, which could improve products and services. For example, being able to model accurately the diffusion of water molecules through biological tissue will advance the interpretation of diffusion-weighted MRI (Magnetic Resonance Imaging) readings. It could also offer more accurate representation of air spreading through food packaging materials, helping to determine shelf life and contamination risk. In addition, quantifying the behavior of foraging animals interacting with macroscopic barriers, such as fences and roads, could provide better predictions on the consequence of climate change for conservation purposes.

    The use of geolocators, mobile phones, and other sensors has seen the tracking revolution generate movement data of ever-increasing quantity and quality over the past 20 years. This has highlighted the need for more sophisticated modelling tools to represent the movement of wide-ranging entities in their environment, from natural organisms to man-made devices.

    Senior author Dr Luca Giuggioli, Associate Professor in Complexity Sciences at the University of Bristol, said: “This new fundamental equation is another example of the importance of constructing tools and techniques to represent diffusion when space is heterogeneous, that is when the underlying environment changes from location to location.

    “It builds on another long-awaited resolution in 2020 of a mathematical conundrum to describe random movement in confined space. This latest discovery is a further significant step forward in improving our understanding of motion in all its shapes and forms – collectively termed the mathematics of movement – which has many exciting potential applications.”

    Science paper:
    Physical Review Research

    See the full article here .

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    Please help promote STEM in your local schools.

    Stem Education Coalition

    The University of Bristol (UK) is one of the most popular and successful universities in the UK and was ranked within the top 50 universities in the world in the QS World University Rankings 2018.

    The U Bristol (UK) is at the cutting edge of global research. We have made innovations in areas ranging from cot death prevention to nanotechnology.

    The University has had a reputation for innovation since its founding in 1876. Our research tackles some of the world’s most pressing issues in areas as diverse as infection and immunity, human rights, climate change, and cryptography and information security.

    The University currently has 40 Fellows of the Royal Society and 15 of the British Academy – a remarkable achievement for a relatively small institution.

    We aim to bring together the best minds in individual fields, and encourage researchers from different disciplines and institutions to work together to find lasting solutions to society’s pressing problems.

    We are involved in numerous international research collaborations and integrate practical experience in our curriculum, so that students work on real-life projects in partnership with business, government and community sectors.

     
  • richardmitnick 4:28 pm on August 9, 2022 Permalink | Reply
    Tags: "U of T Astro SURP Student Spotlight: Louis Branch", , , , , Mathematics, ,   

    From The University of Toronto Dunlap Institute for Astronomy and Astrophysics (CA) : “U of T Astro SURP Student Spotlight: Louis Branch” 

    From The University of Toronto Dunlap Institute for Astronomy and Astrophysics (CA)

    At

    The University of Toronto (CA)

    8.8.22

    1
    Credit: Louis Branch.

    Louis was born in Rio de Janeiro, Brazil, and immigrated to Canada in 2014. He is starting his second year at U of T as a transfer student after several years working as a software developer. He plans to major in Astronomy and Astrophysics.

    Outside of work, Louis is often pondering the implausible chances for most alien species in Star Trek to be humanoids.

    What made you decide to participate in SURP?

    I thought it would be an incredible opportunity to get hands-on experience doing scientific research, while also using some of my programming background to help answer questions in Astronomy.

    After being away from university for so many years, I was worried that my current knowledge of mathematics and physics would present a major roadblock for the research. However, I was wrong. The program and my supervisors have provided an amazing environment for learning, exploring, and asking questions. And I have many questions!

    What is your favourite thing about SURP?

    The best part of my research is having access to measurements made by a six-metre ground telescope.

    The amount of effort and time that goes into specifying, building, and operating such devices is astonishing. It is a humbling experience to use the telescope data knowing that hundreds of people from different continents worked together for several years to make this a possibility. It truly takes a village!

    The seminars are also a highlight of the program. The speakers have covered a wide variety of topics in Astronomy and other academic related themes, such as work/life balance and how to build an effective CV. My favourite one so far was how to give memorable presentations.

    Can you tell us about your research project?

    I am working on the project “Time domain science with Cosmic Microwave Background (CMB) radiation data” under the supervision of Dr. Yilun Guan and Prof. Adam Hincks.

    In this project we are using data from the Atacama Cosmology Telescope in Chile to try to detect pulsars that “glow” in the microwave part of the electromagnetic spectrum.

    The project is exciting for several reasons. First, time domain science with CMB data is a new subfield in astronomy and the work we are doing could pave the way for future projects. The other reason is that neutron stars are usually probed using radio frequencies, so studying them at a different wavelengths could bring new insights on the extreme physical environment inside and surrounding those stars.

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

    Since I restarted university last year, there has been a big gap between what I am currently studying in undergrad and what I have been learning for the project. However, my supervisors have been kind and patient, and will explain (and re-explain) topics that I am not familiar with yet. To bridge this gap is also a source of motivation for myself, especially when things don’t go as planned during the academic semesters.

    What are your plans for the future?

    I would like to work with astronomy education and public outreach. In particular, I am interested in developing interactive tools and visuals to help others without a strong background in mathematics or physics to better understand the Universe and our place in it.

    Such tools could be extremely useful to students from underserved communities without access to a formal education in science and astronomy. This issue is close to my heart because I have always struggled with exact sciences and a career in astronomy would have been nearly impossible if I had stayed in Brazil.

    See the full article here .


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    Please help promote STEM in your local schools.

    Stem Education Coalition

    Dunlap Institute campus

    The Dunlap Institute for Astronomy & Astrophysics (CA) at the 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), 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.

    The University of Toronto participates in the CHIME Canadian Hydrogen Intensity Mapping Experiment at The Canada NRCC Dominion Radio Astrophysical Observatory in Penticton, British Columbia(CA) Altitude 545 m (1,788 ft).


    The 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.

    The 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 outside the United States, the other being McGill University [Université 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 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 10:09 am on August 6, 2022 Permalink | Reply
    Tags: "Mathematically Percolating", "Phase transition": the point where an abrupt qualitive change in the system occurs., , If you look at the connections between the nodes at this phase transition and under different scales you will start to see the fractal and winding math., Mathematicians are interested in the pure math which can be very complex and interesting. In general we draw out grids with edges and nodes where the edges connect the nodes., Mathematics, Percolation theory is a way of describing clustered components in random networks and can be applied to complex things like the spread of an infection through a population., , , The most basic question is to figure out if the phase transition occurs with a jump-what we call discontinuous-or more smoothy-what we call continuous., The two-dimensional models are very well understood and even models with 100 dimensions are easier to understand. But the three- four- and five-dimensional cases are extremely hard to study., This problem has been open for a really long time and needs to be solved before we can move on to understanding all the cool fractal stuff that should be happening at the phase transition.   

    From The California Institute of Technology: “Mathematically Percolating” 

    Caltech Logo

    From The California Institute of Technology

    8.5.22

    1
    Tom Hutchcroft explains why phase transitions in percolation models are so fascinating. Credit: Caltech.

    2
    These images show three different phases in a percolation model: before, at, and beyond the critical phase transition. As the probability of an edge being blue, or p, goes up (from left to right), it becomes easier for the blue portions to spread across the grid. At the point of the phase transition (middle), fractal-like clusters emerge that take long, meandering paths across the grid. Credit: Nils Berglund.

    When water flows through a bed of ground espresso beans, ultimately resulting in a delicious latte, the water is undergoing a process called percolation. The water slowly meanders through the coffee at just the right rate to extract the rich coffee flavors. In general, percolation refers to liquids filtering through a porous medium. The process can describe not only the generation of lattes, but also a host of other phenomena, such as how diseases spread and even physics concepts such as magnetism.

    For mathematicians like Tom Hutchcroft, who joined the Caltech faculty last year as a professor of mathematics, the most interesting aspect of percolation is what happens during a phase transition, the point where an abrupt qualitive change in the system occurs. “You only change one factor in the system a tiny bit, and then you get a big change,” he says. A classical phase transition occurs when water freezes.

    In mathematical percolation models, phase transitions can result in “really interesting mathematical behavior,” according to Hutchcroft, including fractal patterns; fractal refers to self-similar patterns seen at different scales.

    2
    Hutchcroft is studying how the geometry of fractal trees like the one depicted here change when viewed in different dimensions.

    Hutchcroft, who was born and raised in England, earned his bachelor’s degree in mathematics from Cambridge University in 2013 and his PhD in mathematics from the University of British Columbia, Canada, in 2017. He held internships at Microsoft Research Theory Group during his graduate studies, and later completed postdoctoral fellowships at University of Cambridge from 2017 to 2021.

    We met with Hutchcroft over Zoom to learn more about the math of percolation and what he is enjoying about Caltech so far.

    What does percolation have to do with the spread of a disease?

    Percolation theory is a way of describing clustered components in random networks and can be applied to complex things like the spread of an infection through a population. Systems like these have phase transitions. With epidemics, there’s a critical point or phase transition called “R nought,” or R0. This value depends on the average number of people that an infected person infects. When R nought is below 1, the epidemic will die out; when it’s above 1, it will grow exponentially. When R is exactly 1, the infection will die out but very slowly. When you look at this point in models, you tend to have a lot of mathematically interesting behavior. And when you use branching, tree-like models for epidemics, which are a form of a percolation model, you’ll get some interesting fractal geometry in the tree. This has been well understood since the 1990s. But even though you are doing something very simple, you get really mathematically rich objects coming out at the end.

    How do mathematicians study these percolation models?

    While physicists and other scientists may study the statistical physics or statistical mechanics of similar systems as a means to explain the behavior of the components, we mathematicians are interested in the pure math, which can be very complex and interesting. In general, we draw out grids, with edges and nodes, where the edges connect the nodes. These are the percolation models that explain how liquid can flow through a porous media. Imagine that for each edge of this grid, you flip a coin that has a probability “p” of being heads. If the coin comes out heads, you keep the edge, and if it comes out tails, you delete the edge.

    When p is small, or the probability of keeping an edge is small, you will end up with small clusters of connections that are like small islands that don’t connect to anything else. When p is greater than the critical parameter at which a phase transition occurs, called pc [pronounced pee-cee], you will get one big, connected cluster. When p is exactly equal to pc, we expect to get large, fractal-like clusters that permeate across the grid but do so in a zero-density way, with extremely long, tortuous paths.

    So, if this model were explaining coffee percolation, then when p is less than pc, the water would not get through the coffee—it would get stuck in the islands of small clusters. When p is larger than pc, the water would readily flow through. When p is equal to pc, at the phase transition, the water would slowly meander through the grinds, which is what you would want for a good cup of espresso.

    If you look at the connections between the nodes at this phase transition and under different scales you will start to see the fractal and winding math.

    What problems are you working on in this field?

    The two-dimensional models are very well understood and even models with 100 dimensions are easier to understand. But the three- four- and five-dimensional cases are extremely hard to study. One thing I’m working on is trying to crack the three-dimensional problem. The most basic question is to figure out if the phase transition occurs with a jump-what we call discontinuous-or more smoothy-what we call continuous. This problem has been open for a really long time and needs to be solved before we can move on to understanding all the cool fractal stuff that should be happening at the phase transition. I’m also working on other related problems, such as long-range percolation where the probability of two nodes having an edge between them depends on the distance between the nodes. Changing how this probability falls off with the distance has a surprisingly similar effect to changing the dimension of the grid and lets us treat the dimension like a continuous parameter.

    What do you love about working on these math problems?

    A lot of the appeal for me is that it’s fun. When you get a good problem, you get hooked on it. Math is like the king of all puzzle games, but it goes beyond puzzles in that you the solution is very insightful. You not only solve the problem, but you build new conceptual frameworks for understanding other math problems.

    How do you like Caltech so far?

    One of the things that drew me to Caltech was the small class size. It feels less like lecturing and more like doing a seminar where you get to interact with everyone individually. I like the tight community here. Of course, the small size of Caltech can mean less interaction with mathematical peers, but a lot of that is going to be balanced out by the fact that the American Institute of Mathematics is moving its headquarters to Caltech. That’s going to bring a lot more activity here, and people will be passing through regularly.

    I also like the mountains in the Pasadena area, which we don’t have back home in the UK. We’ve been going out hiking. You can look at a mountain on campus and then get in the car and be there in 15 minutes.

    Anything else you’d like to add?

    Since coming to Caltech, I’ve also set up the LA Probability Forum, a monthly mini-conference for the LA-area probability community, so that I get to regularly interact with my colleagues at UCLA and USC, and anyone else who would like to be involved. This has been really enriching for me both scientifically and socially.

    See the full article here .


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    Please help promote STEM in your local schools.

    Stem Education Coalition

    Caltech campus

    The The California Institute of Technology is a private research university in Pasadena, California. The university is known for its strength in science and engineering, and is one among a small group of institutes of technology in the United States which is primarily devoted to the instruction of pure and applied sciences.

    The California Institute of Technology was founded as a preparatory and vocational school by Amos G. Throop in 1891 and began attracting influential scientists such as George Ellery Hale, Arthur Amos Noyes, and Robert Andrews Millikan in the early 20th century. The vocational and preparatory schools were disbanded and spun off in 1910 and the college assumed its present name in 1920. In 1934, The California Institute of Technology was elected to the Association of American Universities, and the antecedents of National Aeronautics and Space Administration ‘s Jet Propulsion Laboratory, which The California Institute of Technology continues to manage and operate, were established between 1936 and 1943 under Theodore von Kármán.

    The California Institute of Technology has six academic divisions with strong emphasis on science and engineering. Its 124-acre (50 ha) primary campus is located approximately 11 mi (18 km) northeast of downtown Los Angeles. First-year students are required to live on campus, and 95% of undergraduates remain in the on-campus House System at The California Institute of Technology. Although The California Institute of Technology has a strong tradition of practical jokes and pranks, student life is governed by an honor code which allows faculty to assign take-home examinations. The The California Institute of Technology Beavers compete in 13 intercollegiate sports in the NCAA Division III’s Southern California Intercollegiate Athletic Conference (SCIAC).

    As of October 2020, there are 76 Nobel laureates who have been affiliated with The California Institute of Technology, including 40 alumni and faculty members (41 prizes, with chemist Linus Pauling being the only individual in history to win two unshared prizes). In addition, 4 Fields Medalists and 6 Turing Award winners have been affiliated with The California Institute of Technology. There are 8 Crafoord Laureates and 56 non-emeritus faculty members (as well as many emeritus faculty members) who have been elected to one of the United States National Academies. Four Chief Scientists of the U.S. Air Force and 71 have won the United States National Medal of Science or Technology. Numerous faculty members are associated with the Howard Hughes Medical Institute as well as National Aeronautics and Space Administration. According to a 2015 Pomona College study, The California Institute of Technology ranked number one in the U.S. for the percentage of its graduates who go on to earn a PhD.

    Research

    The California Institute of Technology is classified among “R1: Doctoral Universities – Very High Research Activity”. Caltech was elected to The Association of American Universities in 1934 and remains a research university with “very high” research activity, primarily in STEM fields. The largest federal agencies contributing to research are National Aeronautics and Space Administration; National Science Foundation; Department of Health and Human Services; Department of Defense, and Department of Energy.

    In 2005, The California Institute of Technology had 739,000 square feet (68,700 m^2) dedicated to research: 330,000 square feet (30,700 m^2) to physical sciences, 163,000 square feet (15,100 m^2) to engineering, and 160,000 square feet (14,900 m^2) to biological sciences.

    In addition to managing NASA-JPL/Caltech , The California Institute of Technology also operates the Caltech Palomar Observatory; the Owens Valley Radio Observatory;the Caltech Submillimeter Observatory; the W. M. Keck Observatory at the Mauna Kea Observatory; the Laser Interferometer Gravitational-Wave Observatory at Livingston, Louisiana and Hanford, Washington; and Kerckhoff Marine Laboratory in Corona del Mar, California. The Institute launched the Kavli Nanoscience Institute at The California Institute of Technology in 2006; the Keck Institute for Space Studies in 2008; and is also the current home for the Einstein Papers Project. The Spitzer Science Center, part of the Infrared Processing and Analysis Center located on The California Institute of Technology campus, is the data analysis and community support center for NASA’s Spitzer Infrared Space Telescope [no longer in service].

    The California Institute of Technology partnered with University of California at Los Angeles to establish a Joint Center for Translational Medicine (UCLA-Caltech JCTM), which conducts experimental research into clinical applications, including the diagnosis and treatment of diseases such as cancer.

    The California Institute of Technology operates several Total Carbon Column Observing Network stations as part of an international collaborative effort of measuring greenhouse gases globally. One station is on campus.

     
  • richardmitnick 11:01 am on July 31, 2022 Permalink | Reply
    Tags: "Hypergraphs Reveal a Solution to a 50-Year-Old Problem", , In 1973 Paul Erdős asked if it was possible to assemble sets of “triples”—three points on a graph—so that they abide by two seemingly incompatible rules., Mathematics,   

    From “WIRED“: “Hypergraphs Reveal a Solution to a 50-Year-Old Problem” 

    From “WIRED“

    Jul 31, 2022
    Leila Sloman

    In 1973 Paul Erdős asked if it was possible to assemble sets of “triples”—three points on a graph—so that they abide by two seemingly incompatible rules.

    1
    Hypergraphs show one possible solution to the so-called schoolgirl problem. Illustration: Samuel Velasco/Quanta Magazine.

    In 1850, Thomas Penyngton Kirkman, a mathematician when he wasn’t fulfilling his main responsibility as a vicar in the Church of England, described his “schoolgirl problem”: “Fifteen young ladies in a school walk out three abreast for seven days in succession: it is required to arrange them daily, so that no two shall walk twice abreast.”

    To a modern mathematician, this kind of problem is best imagined as a hypergraph—a set of nodes collected in groups of three or more. The 15 schoolgirls are nodes, and each group of “three abreast” can be thought of as a triangle, with three lines, or edges, connecting three nodes.

    Kirkman’s problem essentially asks whether there’s an arrangement of these triangles that connects all the schoolgirls to one another, but with the added restriction that no two triangles share an edge. Edge-sharing would imply that two schoolgirls have to walk together more than once. This restriction means each girl walks with two new friends every day for a week, so that every possible pair gets together exactly once.

    This problem and others like it have beguiled mathematicians for the nearly two centuries since Kirkman posed his question. In 1973, the legendary mathematician Paul Erdős posed a similar one. He asked whether it’s possible to build a hypergraph with two seemingly incompatible properties. First, every pair of nodes must be connected by exactly one triangle, as with the schoolgirls. This property makes the graph dense with triangles. The second requirement forces the triangles to be spread out in a very precise way. (Specifically, it requires that for any small group of triangles, there are at least three more nodes than there are triangles.) “You have this slightly contradictory behavior where you have a dense overall object that has no dense parts,” said David Conlon, a mathematician at the California Institute of Technology.

    This January, in an intricate 50-page proof [below], four mathematicians proved that it’s always possible to build such a hypergraph as long as you have enough nodes. “The amount of technicality that they went through just to get this was amazing,” said Allan Lo, a mathematician at the University of Birmingham. Conlon concurred: “It’s a really impressive piece of work.”

    The research team built a system that satisfied Erdős’ devilish requirements by starting with a random process for choosing triangles and engineering it with extreme care to suit their needs. “The number of difficult modifications that go into the proof is actually kind of staggering,” said Conlon.

    Their strategy was to carefully build the hypergraph out of individual triangles. For example, imagine our 15 schoolgirls. Draw a line between each pair.

    2
    All possible connections between 15 nodes. Illustration: Merrill Sherman/Quanta Magazine.

    The goal here is to trace out triangles on top of these lines such that the triangles satisfy two requirements: First, no two triangles share an edge. (Systems that fulfill this requirement are called Steiner triple systems.) And second, ensure that every small subset of triangles utilizes a sufficiently large number of nodes.

    The way the researchers did this is perhaps best understood with an analogy.

    Say that instead of making triangles out of edges, you’re building houses out of Lego bricks. The first few buildings you make are extravagant, with structural reinforcements and elaborate ornamentation. Once you’re done with these, set them aside. They’ll serve as an “absorber”—a kind of structured stockpile.

    Now start making buildings out of your remaining bricks, proceeding without much planning. When your supply of Legos dwindles, you may find yourself with some stray bricks, or homes that are structurally unsound. But since the absorber buildings are so overdone and reinforced, you can pluck some bricks out here and there and use them without courting catastrophe.

    In the case of the Steiner triple system, you’re trying to create triangles. Your absorber, in this case, is a carefully chosen collection of edges. If you find yourself unable to sort the rest of the system into triangles, you can use some of the edges that lead into the absorber. Then, when you’re done doing that, you break down the absorber itself into triangles.

    Absorption doesn’t always work. But mathematicians have tinkered with the process, finding new ways to weasel around obstacles. For example, a powerful variant called iterative absorption divides the edges into a nested sequence of sets, so that each one acts as an absorber for the next biggest.

    “Over the last decade or so there’s been massive improvements,” said Conlon. “It’s something of an art form, but they’ve really carried it up to the level of high art at this point.”

    Erdős’ problem was tricky even with iterative absorption. “It became pretty clear pretty quickly why this problem had not been solved,” said Mehtaab Sawhney, one of the four researchers who solved it, along with Ashwin Sah, who like Sawhney is a graduate student at the Massachusetts Institute of Technology; Michael Simkin, a postdoctoral fellow at the Center of Mathematical Sciences and Applications at Harvard University; and Matthew Kwan, a mathematician at the Institute of Science and Technology Austria. “There were pretty interesting, pretty difficult technical tasks.”

    For example, in other applications of iterative absorption, once you finish covering a set—either with triangles for Steiner triple systems, or with other structures for other problems—you can consider it dealt with and forget about it. Erdős’ conditions, however, prevented the four mathematicians from doing that. A problematic cluster of triangles could easily involve nodes from multiple absorber sets.

    “A triangle you chose 500 steps ago, you need to somehow remember how to think about that,” said Sawhney.

    What the four eventually figured out was that if they chose their triangles carefully, they could circumvent the need to keep track of every little thing. “What it’s better to do is to think about any small set of 100 triangles and guarantee that set of triangles is chosen with the correct probability,” said Sawhney.

    The authors of the new paper [below] are optimistic that their technique can be extended beyond this one problem. They have already applied their strategy to a problem about Latin squares, which are like a simplification of a sudoku puzzle.

    Beyond that, there are several questions that may eventually yield to absorption methods, said Kwan. “There’s so many problems in combinatorics, especially in design theory, where random processes are a really powerful tool.” One such problem, the Ryser-Brualdi-Stein conjecture, is also about Latin squares and has awaited a solution since the 1960s.

    Though absorption may need further development before it can fell that problem, it has come a long way since its inception, said Maya Stein, the deputy director of the Center for Mathematical Modeling at the University of Chile. “That’s something that’s really great to see, how these methods evolve.”

    Science paper:
    An intricate 50-page proof

    See the full article here .

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  • richardmitnick 8:15 pm on July 20, 2022 Permalink | Reply
    Tags: "Mapping the Genome of the Universe", , , , , Mathematics, , , The Flatiron Institute,   

    From The Simons Foundation : “Mapping the Genome of the Universe” 

    From The Simons Foundation

    7.19.22

    One would be forgiven for thinking that the Simons Collaboration on Learning the Universe sounds as absurdly ambitious as a Theory of Everything. But, it turns out, nothing is off the table. “We’d like to understand the big questions,” says collaboration director Greg Bryan, an astronomy professor at Columbia University. “Where did we come from? What is the fate of the universe? Why does it look the way it does?” These questions are huge, to say the least, but scientists in the collaboration believe that they will be able to shed new light on these age-old mysteries in the next few years.

    In the early days of the expansion of the universe, matter was nearly uniformly distributed. The tiny deviations from true uniformity would evolve under forces such as gravity to create areas that were more and less dense, eventually coalescing into the galaxies, planets and stars we see today.

    “If we knew exactly what those initial fluctuations were at that very early time, then we could predict the current universe,” Bryan says. Unfortunately, the beginning of the universe was a long time ago, and no one was there to look around.

    So Learning the Universe researchers must instead deduce the past by observing the current universe and working backward. “We have good observations of the universe now, and going back in time quite a ways,” says collaboration principal investigator Shirley Ho, an astrophysicist at Princeton University and group leader of Cosmology X Data Science at The Flatiron Institute. “It’s like if we have a good video of a person’s life from high school to 30 years old — but we want to use it to figure out their genome.”

    A great deal of the collaboration’s work involves trying out different initial conditions of the matter and energy in the universe, the equivalent of individual genes in a person’s genome, and determining whether our current observations are consistent with a universe that grew up with those ‘genes.’ One of the biggest challenges is the sheer size of the genome they are working with: There are on the order of a million parameters that must be determined. “No one has tried to do such a high-dimensional inference problem before,” Ho says.

    Until that time, standard models had predicted that the gravitational attraction between all matter in the universe should be causing expansion to slow overall. Physicists developed the idea of “dark energy” to explain the acceleration, but what exactly “dark energy” is and how it functions are still unknown.
    ___________________________________________________________________
    The Dark Energy Survey

    Dark Energy Camera [DECam] built at The DOE’s Fermi National Accelerator Laboratory.

    NOIRLab National Optical Astronomy Observatory Cerro Tololo Inter-American Observatory(CL) Victor M Blanco 4m Telescope which houses the Dark-Energy-Camera – DECam at Cerro Tololo, Chile at an altitude of 7200 feet.

    NOIRLabNSF NOIRLab NOAO Cerro Tololo Inter-American Observatory(CL) approximately 80 km to the East of La Serena, Chile, at an altitude of 2200 meters.

    Timeline of the Inflationary Universe WMAP.

    The Dark Energy Survey is an international, collaborative effort to map hundreds of millions of galaxies, detect thousands of supernovae, and find patterns of cosmic structure that will reveal the nature of the mysterious dark energy that is accelerating the expansion of our Universe. The Dark Energy Survey began searching the Southern skies on August 31, 2013.

    According to Albert Einstein’s Theory of General Relativity, gravity should lead to a slowing of the cosmic expansion. Yet, in 1998, two teams of astronomers studying distant supernovae made the remarkable discovery that the expansion of the universe is speeding up.

    To explain cosmic acceleration, cosmologists are faced with two possibilities: either 70% of the universe exists in an exotic form, now called Dark Energy, that exhibits a gravitational force opposite to the attractive gravity of ordinary matter, or General Relativity must be replaced by a new theory of gravity on cosmic scales.

    The Dark Energy Survey is designed to probe the origin of the accelerating universe and help uncover the nature of Dark Energy by measuring the 14-billion-year history of cosmic expansion with high precision. More than 400 scientists from over 25 institutions in the United States, Spain, the United Kingdom, Brazil, Germany, Switzerland, and Australia are working on the project. The collaboration built and is using an extremely sensitive 570-Megapixel digital camera, DECam, mounted on the Blanco 4-meter telescope at Cerro Tololo Inter-American Observatory, high in the Chilean Andes, to carry out the project.

    Over six years (2013-2019), the Dark Energy Survey collaboration used 758 nights of observation to carry out a deep, wide-area survey to record information from 300 million galaxies that are billions of light-years from Earth. The survey imaged 5000 square degrees of the southern sky in five optical filters to obtain detailed information about each galaxy. A fraction of the survey time is used to observe smaller patches of sky roughly once a week to discover and study thousands of supernovae and other astrophysical transients.
    ___________________________________________________________________

    A great deal of the collaboration’s work involves trying out different initial conditions of the matter and energy in the universe, the equivalent of individual genes in a person’s genome, and determining whether our current observations are consistent with a universe that grew up with those ‘genes.’ One of the biggest challenges is the sheer size of the genome they are working with: There are on the order of a million parameters that must be determined. “No one has tried to do such a high-dimensional inference problem before,” Ho says.

    In addition to addressing the distribution of matter in the early universe, the researchers must confront questions about the physical laws governing the early universe in order to understand why it progressed the way it did. The rate of expansion of the universe is one of the core mysteries the collaboration seeks to grasp. Nearly 25 years ago, cosmologists and astrophysicists discovered that not only was the universe expanding, but its expansion was accelerating.

    Saul Perlmutter (center) [The Supernova Cosmology Project] shared the 2006 Shaw Prize in Astronomy, the 2011 Nobel Prize in Physics, and the 2015 Breakthrough Prize in Fundamental Physics with Brian P. Schmidt (right) and Adam Riess (left) [The High-z Supernova Search Team] for providing evidence that the expansion of the universe is accelerating.

    Until that time, standard models had predicted that the gravitational attraction between all matter in the universe should be causing expansion to slow overall. Physicists developed the idea of “dark energy” to explain the acceleration, but what exactly dark energy is and how it functions are still unknown.

    2
    Three simulations of a region of space 480 million light-years across (top row), with zooms of the box marked “A” shown in the bottom row. The first simulation (left column) was run at low resolution; the second (middle column) was carried out at high resolution; the third (right column) used machine learning to augment the low-resolution simulation as if it had been run at high resolution, but at a fraction of the computational cost. Credit:
    Y. Li et al./Proceedings of the National Academy of Sciences 2021

    An undertaking this complex and vast in scope requires input from multiple disciplines. Experts in simulating the evolution of galaxies take initial conditions and create computer models that show how those conditions change over time. But the collaboration needs millions, if not billions, of these simulations, and at the present time, the simulations run much too slowly. That’s where a second group comes in. The machine-learning group is working to speed up the simulations [Y. Lee et al. above], eventually by many orders of magnitude, so that they can perform more simulations with the same amount of computing power. “If we can run a million simulations in the time it used to take one, it means we can try a million different recipes of the genome,” Ho says. Before the Learning the Universe collaboration began, she and her colleagues used convolutional neural networks, the same kind of machine-learning technique used in image recognition software, to speed up simulations of dark matter particles, generating the ‘skeletons’ of the simulated universe. “That step usually took a day or so; now, it takes milliseconds because of what we developed,” she says.

    A third group made up of cosmologists, both theoretical and observational, oversees the comparison of models to real data and helps the collaboration determine which aspects of the observable universe should be compared to aspects of the simulated universes to determine how similar the universes are. The final group in the collaboration consists of statisticians, who develop new techniques to define how probable it is that a simulated model is consistent with our present-day universe. Because of the size of the problem they are working on, traditional techniques that compute a quantity known as the likelihood are intractable; instead, the statisticians are working on developing implicit likelihood inference methods that allow them to get at the relevant probabilities in a different way.

    As the collaboration develops, improvements by each group will circle around to the other groups, improving all models in terms of both accuracy and speed. Although there are challenges inherent in working across fields with different jargon and training, each area of expertise is needed to tackle the collaboration’s ambitious goals. “We all have to bring in our point of view and knowledge and try to figure out how to get this working,” Ho says.

    Collaboration members are realistic: They know that they will not solve all of cosmology in a few years. If they can show that the new tools they are developing are successful for a few data sources and a few specific problems, they and other groups of researchers will be able to apply the same techniques to information coming in from new telescopes, expanding the problems they work on.

    “We want to demonstrate that we can do what we want to do for this subset of cosmological problems,” Bryan says. “Our goal is to show that this is a viable path.”

    See the full article here.

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    Mission and Model

    The Simons Foundation’s mission is to advance the frontiers of research in mathematics and the basic sciences.

    Co-founded in New York City by Jim and Marilyn Simons, the foundation exists to support basic — or discovery-driven — scientific research undertaken in the pursuit of understanding the phenomena of our world.

    The Simons Foundation’s support of science takes two forms: We support research by making grants to individual investigators and their projects through academic institutions, and, with the launch of the Flatiron Institute in 2016, we now conduct scientific research in-house, supporting teams of top computational scientists.

     
  • richardmitnick 12:41 pm on July 7, 2022 Permalink | Reply
    Tags: "Mathematical calculations show that quantum communication across interstellar space should be possible", , If other intelligent beings exist in the Milky Way they could already be trying to communicate with us using such technology and we could begin looking for them., Mathematics, Prior research has shown that it would be nearly impossible to intercept such messages without detection., Prior research has shown that the space between the stars is pretty clean. But is it clean enough for quantum communications?, , Quantum teleportation across interstellar space should be possible., , With quantum communications engineers are faced with quantum particles that lose some or all of their unique characteristics as they interact with obstructions in their path., X-ray photons could travel hundreds of thousands of light years without becoming subject to decoherence.   

    From The University of Edinburgh (SCT) via “phys.org” : “Mathematical calculations show that quantum communication across interstellar space should be possible” 

    From The University of Edinburgh (SCT)

    Via

    “phys.org”

    July 6, 2022
    Bob Yirka

    1
    Credit: Pixabay/CC0 Public Domain.

    A team of physicists at the University of Edinburgh’s School of Physics and Astronomy has used mathematical calculations to show that quantum communications across interstellar space should be possible. In their paper published in the journal Physical Review D, the group describes their calculations and also the possibility of extraterrestrial beings attempting to communicate with us using such signaling.

    Over the past several years, scientists have been investigating the possibility of using quantum communications as a highly secure form of message transmission. Prior research has shown that it would be nearly impossible to intercept such messages without detection. In this new effort, the researchers wondered if similar types of communications might be possible across interstellar space. To find out, they used math that describes that movement of X-rays across a medium, such as those that travel between the stars. More specifically, they looked to see if their calculations could show the degree of decoherence that might occur during such a journey.

    With quantum communications engineers are faced with quantum particles that lose some or all of their unique characteristics as they interact with obstructions in their path—they have been found to be quite delicate, in fact. Such events are known as decoherence, and engineers working to build quantum networks have been devising ways to overcome the problem. Prior research has shown that the space between the stars is pretty clean. But is it clean enough for quantum communications? The math shows that it is. Space is so clean, in fact, that X-ray photons could travel hundreds of thousands of light years without becoming subject to decoherence—and that includes gravitational interference from astrophysical bodies. They noted in their work that optical and microwave bands would work equally well.

    The researchers noted that because quantum communication is possible across the galaxy, if other intelligent beings exist in the Milky Way they could already be trying to communicate with us using such technology and we could begin looking for them. They also suggest that quantum teleportation across interstellar space should be possible.

    See the full article here.

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    The The University of Edinburgh (SCT) , founded in 1582, is the sixth oldest university in the United Kingdom and English-speaking world and one of Scotland’s ancient universities. The university has five main campuses in the city of Edinburgh, which include many buildings of historical and architectural significance such as those in Old Town. The university played an important role in Edinburgh becoming a chief intellectual centre during the Scottish Enlightenment, contributing to the city being nicknamed the “Athens of the North”.

    The university is a member of a number of prestigious academic organizations, including The Russell Group, The Coimbra Group, The Universitas 21, and The League of European Research Universities, a consortium of 23 leading research universities in Europe. It has the third largest endowment of any university in the United Kingdom, after The University of Cambridge (UK) and The University of Oxford. In 2019-20, the university has a consolidated annual income of £1,125.3 million, of which £296.1 million was from research grants and contracts.

    The alumni of the university include some of the major figures of modern history, including three signatories of the United States Declaration of Independence and nine heads of state and government (including three Prime Ministers of the United Kingdom). As of 2020, Edinburgh’s alumni, faculty members and researchers include 19 Nobel laureates; three Turing Award laureates; an Abel Prize winner and Fields Medalist; two Pulitzer Prize winners; two currently sitting UK Supreme Court Justices; and several Olympic gold medalists. It continues to have links to the British Royal Family, having had the Duke of Edinburgh as its Chancellor from 1953 to 2010 and Princess Anne since 2011.

    Edinburgh receives approximately 60,000 applications every year, making it the second most popular university in the United Kingdom by volume of applications. It has the 4th-highest average UCAS entry tariff in Scotland, and 8th overall in the United Kingdom.

    Founding

    Founded by the Edinburgh Town Council, the university began life as a college of law using part of a legacy left by a graduate of the University of St Andrews, Bishop Robert Reid of St Magnus Cathedral, Orkney. Through efforts by the council and ministers of the city, such as John Knox’s successor James Lawson, the institution broadened in scope and became formally established as a college by a Royal Charter, granted by King James VI on 14 April 1582. This was unprecedented in newly Presbyterian Scotland, as older universities in Scotland had been created through Papal bulls. Established as the “Tounis College”, it opened its doors to students in October 1583. Instruction began under the charge of another St Andrews graduate, theologian Robert Rollock. It was the fourth Scottish university in a period when the richer and much more populous England had only two. The school was renamed King James’s College in 1617. By the 18th century, the university was a leading centre of the Scottish Enlightenment.

    2000 to present

    The Edinburgh Cowgate Fire of December 2002 destroyed a number of university buildings, including some 3,000 m^2 (30,000 sq ft.) of the School of Informatics at 80 South Bridge. This was replaced with the Informatics Forum on the central campus, completed in July 2008.

    The Edinburgh Cancer Research Centre (ECRC) was opened in 2002 by The Princess Royal on the Western General Hospital site. In 2007, the MRC Human Genetics Unit formed a partnership with the Centre for Genomic and Experimental Medicine and the Edinburgh Cancer Research Centre to create the Institute of Genetics and Molecular Medicine (IGMM).

    The Euan MacDonald Centre was established in 2007 as a research centre for motor neuron disease (MND). The centre was part funded by a donation from Scottish entrepreneur Euan MacDonald and his father Donald.

    On 1 August 2011, the Edinburgh College of Art (founded in 1760) merged with the university’s School of Arts, Culture and Environment.

    The Scottish Centre for Regenerative Medicine, a stem cell research centre dedicated to the development of regenerative treatments, was opened by the Anne, Princess Royal on 28 May 2012. It is home to biologists and clinical academics from the MRC Centre for Regenerative Medicine (CRM), and applied scientists working with the Scottish National Blood Transfusion Service and Roslin Cells. On 25 August 2014, the centre reported on the first working organ, a thymus, grown from scratch inside an animal.

    In 2014, the Zhejiang University-University of Edinburgh Institute was founded as a joint institute offering degrees in biomedical sciences, taught in English. The campus, located in Haining, Zhejiang Province, China, opened on 16 August 2016.

    Beginning in 2015, the University of Edinburgh maintains a Wikimedian in Residence.

    In 2018, the University of Edinburgh was a signatory in the landmark £1.3bn Edinburgh and South East Scotland City Region Deal, with the UK and Scottish governments, six local authorities and all universities and colleges in the region. The University committed to delivering a range of economic benefits to the region through the Data-Driven Innovation initiative. In conjunction with Heriot-Watt University (SCT), the initiative created four innovation hubs – the Bayes Centre; Usher Institute; Edinburgh Futures Institute; Easter Bush Campus; and one based at Heriot-Watt, the National Robotarium. The deal also included creation of the Edinburgh International Data Facility, which performs high-speed data processing in a secure environment.

     
  • richardmitnick 10:46 am on May 31, 2022 Permalink | Reply
    Tags: "Simple questions with difficult answers", , Mathematics, Oliver Janzer is a mathematician with heart and soul.,   

    From The Swiss Federal Institute of Technology in Zürich [ETH Zürich] [Eidgenössische Technische Hochschule Zürich] (CH): “Simple questions with difficult answers” 

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

    31.05.2022
    Barbara Vonarburg

    1
    Oliver Janzer is a mathematician with heart and soul. He particularly enjoys the idea that a conclusive proof in pure mathematics will still be valid 100 or 1,000 years from now. (Photo: ETH Zürich / Alessandro Della Bella)

    Oliver Janzer is a mathematician who specialises in graphs – that is, collections of nodes that may or may not be connected, such as Facebook users. As part of his ETH Fellowship, the young researcher has been busy solving problems that had stumped mathematicians for decades.

    Imagine a group of people. Some of them know each other; others have never met. If you connect all the pairs of individuals who know each other, while making no connections between strangers, you end up with a network of nodes and interconnections. Mathematicians refer to such a network as a graph. Social networks like Facebook can be considered as a graph. The theory of these networks is a subfield of combinatorics, a field of mathematics in which Oliver Janzer specialises. The 27-​year-old scientist has been working as an ETH Fellow in the group of mathematics professor Benny Sudakov since the autumn of 2020.

    “We tackle specific kinds of problems in graph theory,” Janzer explains. This might mean calculating the maximum number of people who can be connected to each other when certain patterns are forbidden – for example, if nobody in the group is permitted to know everyone else. “The questions we ask in our field tend to be fairly simple compared to the ones you get in other fields of mathematics,” Janzer says, “but somehow the solutions are not always very simple!” This is a huge understatement by the young researcher, who has already won several awards. One maths problem first formulated in 1975 had resisted every attempt to find a solution – but Janzer and Sudakov recently came up with new ideas that gave them the breakthrough they needed to solve it.

    Translated from mathematical jargon into everyday English, the problem basically concerned the following question: How many pairs of individuals can know each other in a group of n people if there are no subgroups in which each person knows exactly k other people, where n is an integer and k is greater than or equal to the number three. “When explaining this problem, I had to cheat a tiny bit to make it easier to grasp,” Janzer confesses. Nonetheless, the solution he and Sudakov came up with is exact – and it generated tremendous interest in the scientific community when it was published as a preprint in late April 2022.

    From Budapest to Cambridge

    Janzer grew up in Budapest, and he acknowledges how fortunate he was to attend a school in Hungary where mathematics teaching has a high priority. “My classmates had similar interests to mine. Maths was our favorite subject and we had some amazing teachers,” he says. It wasn’t long before he was making waves in a national mathematics competition and, once he hit his teens, he went on to win a bronze and two silver medals at the International Mathematical Olympiad. This paved the way for him to study mathematics at Cambridge University in the UK. “That was when things really changed,” Janzer says: “Maths at university was very different to what I was used to at school and in the competitions.”

    Janzer continued to take great pleasure in the subject, even when the time came to tackle his doctoral project in Cambridge: “That was another big shift, because suddenly it wasn’t just about learning things, but about doing my own research!” Before long, he was producing publishable results that were attracting the interest of experts, including ETH Professor Sudakov. “He sent me an e-​mail asking what my plans were after I finished my doctorate,” Janzer says. With Sudakov on board as his mentor, he decided to apply for an ETH Zürich Postdoctoral Fellowship, a programme that aims to support up-​and-coming researchers who achieve outstanding results early on in their career. “I was very happy that it all worked out,” Janzer says, because the success rate for applicants is only about 25 percent.

    Cash prize for disproving a conjecture

    The Hungarian scientist Paul Erdős, one of the most important mathematicians of the 20th century, probably has played a key role in Janzer’s career so far. Erdős posed numerous questions and conjectures in combinatorics, one of which Janzer was able to disprove last year. “That’s another piece of work I’m really proud of!” he says. Erdős had even set up a prize fund for anyone who solved the problem he formulated in 1981, with 250 dollars awarded for proof of his conjecture and 500 dollars for its refutation.

    The discoveries in Janzer’s specialist area are mainly applicable in the field of mathematics itself; real-​world applications tend to be few and far between, though there are exceptions. “Imagine you’re drawing up the match schedule for a football league, for example,” Janzer says: “If the league has 20 teams, it is unlikely that just by hand you can design a suitable schedule. Fortunately, our work provides some useful techniques to get the job done.” Many of the applications that do emerge, he says, aren’t for specific results, but for the underlying ideas. He cites the example of randomised techniques, which were developed to solve a purely combinatorial problem; these led to randomised algorithms, which are ubiquitous in modern computer science. The results of the latest work by Janzer and Sudakov can also be applied in this field, specifically in regard to machine learning.

    “On the whole, though, it’s rare for our studies to be motivated by applications,” Janzer says. The value of a piece of work is generally determined by how well it advances the research itself, how old the problem is, and how many other scientists have already attempted to tackle it. “I sometimes think it would be nice to see my research make an impact in the real world, but I am well compensated by other aspects!” Janzer says, emphasising just how beautiful his discipline is. He particularly enjoys the idea that a conclusive proof in pure mathematics will still be valid 100 or 1,000 years from now. “It’s nice to think of your own name always being associated with that theorem.”

    All Janzer needs for his work is a pen and paper, a computer is only involved to read and write scientific articles. On average, some 15 new publications a day appear in his subject area, and he often starts browsing through them over breakfast. Once he gets to his office on the top floor of the ETH Main Building, he continues his perusal of the literature and sets to work solving mathematical problems. “In many cases, the only way to make progress is to adopt a new perspective,” he says. “And sometimes it helps to ask a slightly different question, whether that’s an easier, more difficult or simply more general one.” The seven members of the research group don’t limit their discussions to mathematics, but also talk about what’s happening in the world outside. “I’m interested in politics and keep up with the latest news,” Janzer says: “Hungary shares a border with Ukraine, and this war is really very sad.”
    ===
    Plenty of sport – plus the odd computer game

    Janzer does plenty of sport to balance out the mental demands of his work. “I enjoy running; it helps me relax, and it is obviously very healthy,” he says. “And I like watching movies and chatting with friends.” He has also become more comfortable talking about his interest in computer games after seeing in a newspaper article that Terence Tao – one of the best-​known mathematicians of our day – spent lots of time playing the computer game Civilization when he was at university. “I was pleased to see that I’m not the only one! You can still be an outstanding mathematician playing this game,” he says.

    Once his ETH Fellowship comes to an end in August, Janzer will be returning to Cambridge to take up another research position, this time for four years. After that, he says he’s open to anything. His younger brother is also studying mathematics at Cambridge and working on a doctoral project in the field of combinatorics, and his sister moved from Cambridge to Oxford to do a Master’s degree in computer science. “For some reason, all three of us chose to pursue mathematics, even though our parents had no connection to the world of mathematics at all!” Janzer says. “That is pretty surprising.”

    See the full article here .

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    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, Stanford University 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, Stanford University, California Institute of Technology, Princeton University, 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 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 .

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

    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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    Please help promote STEM in your local schools.

    Stem Education Coalition

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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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    Please help promote STEM in your local schools.

    Stem Education Coalition

    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.

     
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