Caltech News tagged with "mathematics"
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enPhysics and Mathematics Professors Named Simons Investigators
http://www.caltech.edu/news/physics-and-mathematics-professors-named-simons-investigators-51251
<div class="field field-name-news-writer field-type-ds field-label-inline clearfix"><div class="field-label">News Writer: </div><div class="field-items"><div class="field-item even">Lori Dajose</div></div></div><div class="field field-name-field-images field-type-file field-label-hidden"><div class="field-items"><div class="field-item even"><div class="ds-1col file file-image file-image-jpeg view-mode-full_grid_9 clearfix ">
<img src="http://s3-us-west-1.amazonaws.com/www-prod-storage.cloud.caltech.edu/styles/article_photo/s3/SimonsInvestigators-2016.jpg?itok=h99I9I_0" alt="" /><div class="field field-name-field-caption field-type-text field-label-hidden"><div class="field-items"><div class="field-item even">Anton Kapustin and Vladimir Markovic.</div></div></div><div class="field field-name-credit-sane-label field-type-ds field-label-hidden"><div class="field-items"><div class="field-item even">Credit: Caltech</div></div></div></div></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><a href="http://pma.caltech.edu/content/anton-n-kapustin">Anton Kapustin</a> (PhD '97), the Earle C. Anthony Professor of Theoretical Physics and Mathematics, and <a href="http://pma.caltech.edu/content/vladimir-markovic">Vladimir Markovic</a>, the John D. MacArthur Professor of Mathematics, have been named Simons Investigators. These appointments are given annually to "support outstanding scientists in their most productive years, when they are establishing creative new research directions," according to the Simons Foundation, which grants the awards. Investigators receive $100,000 annually for five years.</p><p>Kapustin studies mathematical physics, particularly dualities—relations between two superficially very different models of quantum fields, which help scientists study the behavior of strongly interacting elementary particles.</p><p>"Recently, my research has focused on the classification of exotic states of quantum matter," Kapustin says. "Such states have been proposed to be useful for building a quantum computer. Surprisingly, it turns out that the classification problem can be attacked using methods of topology, a branch of geometry which studies properties of geometric shapes which are not affected by continuous deformations."</p><p>Markovic focuses on various aspects of low-dimensional geometry, which is the study of shapes and forms that certain topological spaces can take.</p><p>"The main themes of my research are manifolds—a particular kind of topological space—and more generally groups, and their geometric, topological and dynamical properties," says Markovic. "Beside this, I have been very interested in certain partial differential equations and geometric flows including harmonic mappings and heat flows."</p><p>"I am excited to be named Simons Investigator," he adds. "This award will enable me to have more time to focus on my research, learn new fields, and test and develop my mathematical ideas."</p></div></div></div><div class="field field-name-field-pr-links field-type-link-field field-label-above"><div class="field-label">Related Links: </div><div class="field-items"><div class="field-item even"><a href="http://www.caltech.edu/news/two-caltech-professors-named-simons-investigators-47457" class="pr-link">Two Caltech Professors Named Simons Investigators</a></div></div></div>Thu, 07 Jul 2016 04:18:44 +0000ldajose51251 at http://www.caltech.eduShou Receives Fellowship for Graduate Studies in Germany
http://www.caltech.edu/news/shou-receives-fellowship-graduate-studies-germany-50934
<div class="field field-name-news-writer field-type-ds field-label-inline clearfix"><div class="field-label">News Writer: </div><div class="field-items"><div class="field-item even">Lori Dajose</div></div></div><div class="field field-name-field-images field-type-file field-label-hidden"><div class="field-items"><div class="field-item even"><div class="ds-1col file file-image file-image-jpeg view-mode-full_grid_9 clearfix ">
<img src="http://s3-us-west-1.amazonaws.com/www-prod-storage.cloud.caltech.edu/styles/article_photo/s3/LShou-DAAD-NEWS-WEB.jpg?itok=idCH5amT" alt="" /><div class="field field-name-field-caption field-type-text field-label-hidden"><div class="field-items"><div class="field-item even">Laura Shou</div></div></div><div class="field field-name-credit-sane-label field-type-ds field-label-hidden"><div class="field-items"><div class="field-item even">Credit: Courtesy of L. Shou</div></div></div></div></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>Laura Shou, a senior in mathematics, has received a Graduate Study Scholarship from the German Academic Exchange Service (DAAD) to pursue a master's degree in Germany. She will spend one year at the Ludwig-Maximilians-Universität München and the Technische Universität München, studying in the theoretical and mathematical physics (TMP) program.</p><p>The DAAD is the German national agency for the support of international academic cooperation. The organization aims to promote international academic relations and cooperation by offering mobility programs for students, faculty, and administrators and others in the higher education realm. The Graduate Study Scholarship supports highly qualified American and Canadian students with an opportunity to conduct independent research or complete a full master's degree in Germany. Master's scholarships are granted for 12 months and are eligible for up to a one-year extension in the case of two-year master's programs. Recipients receive a living stipend, health insurance, educational costs, and travel.</p><p>"As a math major, I was especially interested in the TMP course because of its focus on the interplay between theoretical physics and mathematics," Shou says. "I would like to use mathematical rigor and analysis to work on problems motivated by physics. The TMP course at the LMU/TUM is one of the few programs focused specifically on mathematical physics. There are many people doing research in mathematical physics there, and the program also regularly offers mathematically rigorous physics classes."</p><p>At Caltech, Shou has participated in the Summer Undergraduate Research Fellowship (SURF) program three times, conducting research with Professor of Mathematics Yi Ni on knot theory and topology, with former postdoctoral fellow Chris Marx (PhD '12) on mathematical physics, and with Professor of Mathematics Nets Katz on analysis. She was the president of the Dance Dance Revolution Club and a member of the Caltech NERF Club and the Caltech Math Club.</p><p>Following her year in Germany, Shou will begin the mathematics PhD program at Princeton.</p></div></div></div>Tue, 07 Jun 2016 18:04:59 +0000ldajose50934 at http://www.caltech.eduOoguri Receives Chunichi Award
http://www.caltech.edu/news/ooguri-receives-chunichi-award-50716
<div class="field field-name-news-writer field-type-ds field-label-inline clearfix"><div class="field-label">News Writer: </div><div class="field-items"><div class="field-item even">Robert Perkins</div></div></div><div class="field field-name-field-images field-type-file field-label-hidden"><div class="field-items"><div class="field-item even"><div class="ds-1col file file-image file-image-jpeg view-mode-full_grid_9 clearfix ">
<img src="http://s3-us-west-1.amazonaws.com/www-prod-storage.cloud.caltech.edu/styles/article_photo/s3/Ooguri_Hirosi.jpeg?itok=lNXz9oTK" alt="" /><div class="field field-name-field-caption field-type-text field-label-hidden"><div class="field-items"><div class="field-item even">Hirosi Ooguri</div></div></div><div class="field field-name-credit-sane-label field-type-ds field-label-hidden"><div class="field-items"><div class="field-item even">Credit: Bill Youngblood for Caltech</div></div></div></div></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><a href="https://www.pma.caltech.edu/content/hiroshi-hirosi-ooguri-oguri">Hirosi Ooguri</a>, the Fred Kavli Professor of Theoretical Physics and Mathematics and founding director of the Walter Burke Institute for Theoretical Physics, will be the 2016 recipient of the Chunichi Cultural Award. Founded in 1947 by Japanese newspaper Chunichi Shimbun to commemorate the enacting of the Japanese constitution, the award celebrates individuals or organizations who have made significant contributions to the arts, humanities, and natural or social sciences. Other awardees this year include physicist and 2015 Nobel Laureate Takaaki Kajita, poet Toru Kitagawa, and biologist Ikue Mori, each of whom will receive the 2 million yen ($20,000) prize. Previous recipients include six other Nobel laureates and one Fields medalist.</p><p>The prize honors Ooguri for the "development of innovative methods of modern mathematics in high energy theory," according to the prize citation. His research focuses on creating new theoretical tools in quantum field theory and superstring theory, which may ultimately lead to a unified theory of the forces and matter in nature. He is particularly renowned for his work on topological string theory, which has had broad applications ranging from black hole physics to algebraic geometry and knot theory in mathematics.</p><p>This April, Ooguri was elected as a fellow of the <a href="/news/american-academy-arts-and-sciences-elects-two-caltech-50547">American Academy of Arts and Sciences</a>. He is also the recipient of the <a href="http://www.caltech.edu/content/physicist-hirosi-ooguri-awarded-novel-research-black-holes">Leonard Eisenbud Prize for Mathematics and Physics</a> from the American Mathematical Society, the Nishina Memorial Prize, the Humboldt Research Award, the <a href="/news/two-caltech-professors-named-simons-investigators-47457">Simons Investigator Award</a>, and is a <a href="http://www.caltech.edu/content/caltech-faculty-named-ams-fellows">fellow of the American Mathematical Society</a>. He also received Japan's <a href="/news/superstring-theorist-honored-science-writing-prize-43479">Kodansha Prize for Science Books</a> for his popular Introduction to Superstring Theory in 2014.</p><p>Ooguri will receive the Chunichi Award at a ceremony to be held in Japan on June 3.</p></div></div></div>Wed, 11 May 2016 19:14:51 +0000abenter50716 at http://www.caltech.eduTom M. Apostol, 1923–2016
http://www.caltech.edu/news/tom-m-apostol-1923-2016-50698
<div class="field field-name-news-writer field-type-ds field-label-inline clearfix"><div class="field-label">News Writer: </div><div class="field-items"><div class="field-item even">Lori Dajose</div></div></div><div class="field field-name-field-images field-type-file field-label-hidden"><div class="field-items"><div class="field-item even"><div class="ds-1col file file-image file-image-jpeg view-mode-full_grid_9 clearfix ">
<img src="http://s3-us-west-1.amazonaws.com/www-prod-storage.cloud.caltech.edu/styles/article_photo/s3/Apostol_Tom_2013.jpg?itok=nPx20o1A" alt="" /></div></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>Tom M. Apostol, professor of mathematics, emeritus, passed away on May 8, 2016. He was 92.</p><p>Apostol was born in Helper, Utah, on August 20, 1923. He received his bachelor of science in chemical engineering in 1944 and a master's degree in mathematics in 1946, both from the University of Washington, Seattle. In 1948, he received his PhD in mathematics from the University of California, Berkeley. In 1950, he arrived at Caltech as an assistant professor; he was named associate professor in 1956, professor in 1962, and professor emeritus in 1992.</p><p>Apostol was the author of several influential textbooks. For more than five decades, undergraduate introductory mathematics courses at Caltech have used Apostol's two-volume text, "<em>Calculus</em>," which is often referred to by Caltech students as "Tommy 1" and "Tommy 2." These volumes, as well as many of his other textbooks in mathematical analysis and analytic number theory, have been translated into Greek, Italian, Spanish, Farsi, and Portuguese. Apostol also worked with a Caltech team that produced <em>The Mechanical Universe . . . and Beyond</em>, a 52-episode video lecture series based on <em>The Mechanical Universe: Introduction to Heat and Mechanics</em> and <em>Beyond the Mechanical Universe: From Electricity to Modern Physics</em>, the introductory physics textbooks that Apostol coauthored.</p><p>Apostol later was the creator, director, and producer of <em>Project MATHEMATICS!</em>, a series of award-winning computer animated videos that explore basic topics in high school mathematics such as the Pythagorean Theorem, scaling, sines and cosines, and the history of mathematics. The nine videos, which are still available for order through the Caltech bookstore, are estimated to have been viewed by 10 million people worldwide.</p><p>"Tom was a great scholar and a beloved teacher and mentor. Generations of Caltech students benefited from his passion and dedication," says Fiona Harrison, the Benjamin M. Rosen Professor of Physics and the Kent and Joyce Kresa Leadership Chair of the Division of Physics, Mathematics and Astronomy.</p><p>"Tom Apostol was a great human being and mathematician, and an inspiration to many. He was very famous the world over for his immense talent for mathematical exposition," says Dinakar Ramakrishnan, Caltech's Taussky-Todd-Lonergan Professor of Mathematics and executive officer for mathematics. "His books set a high standard but remained accessible to many, as decades of Caltech undergraduates would testify, while his videos have stimulated high school students to pursue the beauty of mathematics."</p><p>In 1982, Apostol received an award for teaching excellence from the Associated Students of the California Institute of Technology (ASCIT). In 1998 the Mathematical Association of America (MAA) awarded him the annual Trevor Evans Award, presented to authors of an exceptional article that is accessible to undergraduates, for his piece entitled "What Is the Most Surprising Result in Mathematics?" (Answer: the prime number theorem). In 2005, 2008, and 2010, he was awarded MMA's Lester R. Ford Award, given to recognize authors of articles of expository excellence. He additionally served as a visiting lecturer for the MMA and as a member of hits Board of Governors.</p><p>He was <a href="http://www.caltech.edu/news/caltech-faculty-named-ams-fellows-37310">named</a> as one of the inaugural class of Fellows of the American Mathematical Society in 2012.</p><p>Apostol, who was an American of Greek descent, spent four months in Greece as a visiting professor of mathematics at the University of Patras in 1978. Additionally, he spent eight years as a member of an Electoral Committee selecting faculty for the University of Crete. In 2001, he was elected as a corresponding member of the Academy of Athens.</p><p>Apostol is survived by his wife, Jane Apostol; his stepson, Stephen Goddard; his sisters, Kay Navrides and Betsie Strouzas; and his brother, John Apostol.</p><p>A memorial service is being planned for later this year.</p></div></div></div>Mon, 09 May 2016 22:04:33 +0000abenter50698 at http://www.caltech.eduSimon Receives Lifetime Achievement Award
http://www.caltech.edu/news/simon-receives-lifetime-achievement-award-48745
<div class="field field-name-news-writer field-type-ds field-label-inline clearfix"><div class="field-label">News Writer: </div><div class="field-items"><div class="field-item even">Kathy Svitil</div></div></div><div class="field field-name-field-images field-type-file field-label-hidden"><div class="field-items"><div class="field-item even"><div class="ds-1col file file-image file-image-jpeg view-mode-full_grid_9 clearfix ">
<img src="http://s3-us-west-1.amazonaws.com/www-prod-storage.cloud.caltech.edu/styles/article_photo/s3/BSimon_059-AS-NEWS-WEB%5B1%5D.jpg?itok=zUgFFThf" alt="" /><div class="field field-name-field-caption field-type-text field-label-hidden"><div class="field-items"><div class="field-item even">Barry Simon</div></div></div><div class="field field-name-credit-sane-label field-type-ds field-label-hidden"><div class="field-items"><div class="field-item even">Credit: Bob Paz</div></div></div></div></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>Barry M. Simon, the International Business Machines (IBM) Professor of Mathematics and Theoretical Physics at Caltech, has been awarded the 2016 Leroy Steele Prize for Lifetime Achievement of the American Mathematical Society (AMS) for his "tremendous impact on the education and research of a whole generation of mathematical scientists through his significant research achievements, highly influential books, and mentoring of graduate students and postdocs," according to the prize citation.</p><p>In conferring the award, the AMS noted Simon's "career of exceptional achievement," which includes the publication of 333 papers and 16 books. Simon was specifically recognized for proving a number of fundamental results in statistical mechanics and for contributing to the construction of quantum fields in two space‐time dimensions—topics that, the AMS notes, have "grown into major industries"—as well as for his "definitive results" on the general theory of Schrödinger operators, work that is crucial to an understanding of quantum mechanics and that has led to diverse applications, from probability theory to theoretical physics. He has also made fundamental contributions to the theory of orthogonal polynomials and their asymptotics.</p><p>"Barry Simon is a powerhouse in mathematical physics and has had an outstanding career which this award attests to," says Vladimir Markovic, the John D. MacArthur Professor of Mathematics. "Caltech is lucky to have him."</p><p>"Barry is a driving force in mathematics at Caltech and has had enormous influence as a scholar, a teacher, and a mentor," says Fiona Harrison, the Benjamin M. Rosen Professor of Physics and holder of the Kent and Joyce Kresa Leadership Chair for the Division of Physics, Mathematics and Astronomy.</p><p>Simon spoke at the International Congress of Mathematics in 1974 and has since given almost every prestigious lecture available in mathematics and physics. He was named a fellow of the American Academy of Arts and Sciences in 2005, and was among the inaugural class of AMS fellows in 2012. In 2015, Simon was awarded the <a href="http://www.caltech.edu/news/simon-wins-international-mathematics-prize-46655">International János Bolyai Prize of Mathematics</a> by the Hungarian Academy of Sciences, given every five years to honor internationally outstanding works in mathematics, and in 2012, he was given <a href="http://www.caltech.edu/news/caltech-professor-barry-simon-wins-henri-poincare-prize-23607">the Henri Poincaré Prize</a> by the International Association of Mathematical Physics. The prize is awarded every three years in recognition of outstanding contributions in mathematical physics and accomplishments leading to novel developments in the field.</p><p>Simon received his AB from Harvard College in 1966 and his doctorate in physics from Princeton University in 1970. He held a joint appointment in the mathematics and physics departments at Princeton for the next decade. He first arrived at Caltech as a Sherman Fairchild Distinguished Visiting Scholar in 1980 and joined the faculty permanently in 1981. He became the IBM Professor in 1984.</p></div></div></div>Thu, 12 Nov 2015 20:25:39 +0000schabner48745 at http://www.caltech.eduKatz Receives Prestigious Award for Mathematics
http://www.caltech.edu/news/katz-receives-prestigious-award-mathematics-47197
<div class="field field-name-news-writer field-type-ds field-label-inline clearfix"><div class="field-label">News Writer: </div><div class="field-items"><div class="field-item even">Lori Dajose</div></div></div><div class="field field-name-field-images field-type-file field-label-hidden"><div class="field-items"><div class="field-item even"><div class="ds-1col file file-image file-image-jpeg view-mode-full_grid_9 clearfix ">
<img src="http://s3-us-west-1.amazonaws.com/www-prod-storage.cloud.caltech.edu/styles/article_photo/s3/NKatz_7578-NEWS-WEB%5B1%5D.jpg?itok=KRBzC2lo" alt="" /></div></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>Caltech professor of mathematics <a href="http://pma.caltech.edu/content/nets-h-katz">Nets Katz</a> has received the 2015 Clay Research Award from the Clay Mathematics Institute. The award was given jointly to Katz and his collaborator, MIT professor of mathematics Larry Guth, for their solution of the Erdős distance problem and for "other joint and separate contributions to combinatorial incidence geometry."</p><p>Combinatorial incidence geometry is the study of possible configurations, or arrangements, between geometric objects such as points or lines. One basic open problem in this field is the Erdős distance problem, for which Katz received the Clay award. The Erdős distance problem examines a set "large" number of points distributed in various arrangements in a two-dimensional plane. In some configurations, like a lattice or grid, the points are evenly spaced. In others, as in a random distribution of points, the spacing between points is varied. The problem asks how many times the same distance can occur between these points, and what is the minimum number of distinct distances possible between these points.</p><p>In 2010, Guth and Katz proved that the minimum number of unique distances between <em>n</em> points, regardless of their spatial configuration, is the number of points <em>n</em> divided by the logarithm of <em>n</em>: <em>n/log(n).</em></p><p>Katz's work on the Erdős problem is an example of his larger research interest in coincidences. By demonstrating that there is a minimum number of unique distances between points, even when in a uniform arrangement like a lattice, Katz showed that coincidences—such as many sets of points having the same distance between them—can occur only a limited number of times.</p><p>Katz received his PhD from the University of Pennsylvania and was a professor of mathematics at Indiana University Bloomington before joining Caltech's faculty in 2013. He was named a Guggenheim Fellow in 2012. Previously, his research was in harmonic analysis, a field concerned with representing functions as superpositions of basic oscillating mathematical "waves."</p><p>The Clay Mathematics Institute is a private foundation "dedicated to increasing and disseminating mathematical knowledge." Given annually, the Clay Research Award recognizes contemporary mathematical breakthroughs.</p></div></div></div>Thu, 02 Jul 2015 14:03:10 +0000schabner47197 at http://www.caltech.eduPrime Numbers, Quantum Fields, and Donuts: An Interview with Xinwen Zhu
http://www.caltech.edu/news/prime-numbers-quantum-fields-and-donuts-interview-xinwen-zhu-45000
<div class="field field-name-news-writer field-type-ds field-label-inline clearfix"><div class="field-label">News Writer: </div><div class="field-items"><div class="field-item even">Jessica Stoller-Conrad</div></div></div><div class="field field-name-field-images field-type-file field-label-hidden"><div class="field-items"><div class="field-item even"><div class="ds-1col file file-image file-image-jpeg view-mode-full_grid_9 clearfix ">
<img src="http://s3-us-west-1.amazonaws.com/www-prod-storage.cloud.caltech.edu/styles/article_photo/s3/Xinwen_Zhu_6301-CC-NEWS-WEB.jpg?itok=z0gPGPZQ" alt="" /><div class="field field-name-field-caption field-type-text field-label-hidden"><div class="field-items"><div class="field-item even">Xinwen Zhu, associate professor of mathematics</div></div></div><div class="field field-name-credit-sane-label field-type-ds field-label-hidden"><div class="field-items"><div class="field-item even">Credit: Lance Hayashida/Caltech Office of Strategic Communications</div></div></div></div></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>In 1994, British mathematician Andrew Wiles successfully developed a proof for Fermat's last theorem—a proof that was once partially scribbled in a book margin by 17th-century mathematician Pierre de Fermat but subsequently eluded even the best minds for more than 300 years. Wiles's hard-won success came after digging into a vast web of mathematical conjectures called the Langlands program. The Langlands program, proposed by Canadian mathematician Robert Phelan Langlands in the 1960s, acts as a bridge between seemingly unrelated disciplines in mathematics, such as number theory—the study of prime numbers and other integers—and more visual disciplines such as geometry. </em></p><p><em>However, to get the ideas he needed for his history-making proof, Wiles only scratched the surface of the Langlands program. Now Xinwen Zhu, an associate professor of mathematics at Caltech, is digging deeper, looking for further applications of this so-called unifying theory of mathematics—and how it can be used to relate number theory to disciplines ranging from quantum physics to the study of donut-shaped geometric surfaces. </em></p><p><em>Zhu came to Caltech from Northwestern University in September. Originally from Sichuan, China, he received his bachelor's degree from Peking University in 2004 and his doctorate from UC Berkeley in 2009. </em></p><p><em>He recently spoke with us about his work, the average day of a mathematician, and his new life in California.</em></p><p> </p><p><strong>Can you give us a general description of your research?</strong></p><p>My work is in mathematics, related to what's called Langlands program. It's one of the most intrinsic parts of modern mathematics. It relates number theory—specifically the study of prime numbers like 2, 3, 5, 7, and so on—to topics as diverse as geometry and quantum physics.</p><p> </p><p><strong>Why do you want to connect number theory to geometry and quantum physics?</strong></p><p>Compared to number theory, geometry is more intuitive. You can see a shape and understand the mathematics that are involved in making that shape. Number theory is just numbers—in this case, just prime numbers. But if we combine the two, then instead of thinking about the primes as numbers, we can visualize them as points on a Riemann surface—a geometric surface kind of represented by the shape of a donut—and the points can move continuously. Think of an ant on a donut—the ant can move freely on the surface. This means that a point on the donut has some intrinsic connections with the points nearby. In number theory it is very difficult to say that any relationship exists between two primes, say 5 and 7, because there are no other primes between them, but there <em>are</em> points between any two points. It is still very difficult to envision, but it gives us a more intuitive way to think about the numbers.</p><p>We want to understand certain things about prime numbers—for example, the distribution of primes among all natural numbers. But that's difficult when you're working with just the numbers; there are very few rules, and everything is unpredictable. The geometric theory here adds a sort of geometric intuition, and the application to quantum field theory adds a physical intuition. Thinking about the numbers and equations in these contexts can give us new insights. I really don't understand exactly how physicists think, but physicists are very smart because they have this intuition. It's just sort of their nature. They can always make the right guess or conjecture. So our hope is to use this sort of intuition to come back to understand what happens in number theory.</p><p> </p><p><strong>Mathematicians don't really have lab spaces or equipment for experiments, so what does a day at the office look like for you? </strong></p><p>Usually I just think. And unfortunately, it's usually without any result, but that's fine. Then, after months and months, one day there is an idea. And that's how we do math. We read papers sometimes to keep our eyes on what the newest development is, but it's probably not as important as it is for other disciplines. Of course, one can also get new ideas and stimulation from reading, so we keep our eyes on what's going on this week.</p><p> </p><p><strong>A two-part question: How did you get first get interested in math in general, and how did you get interested in this particular field that you're in now?</strong></p><p>My interest in math began when I was a child. People can usually count numbers at a pretty early age, but I was interested in math and could do calculations a little bit quicker and a bit younger than others. It came naturally to me. Also, my grandfather was a chemist and physicist, and he always emphasized the importance of math.<br /><br />But to be honest, I didn't really know anything about this aspect of the Langlands program until I was in graduate school at Berkeley. My adviser, Edward Frenkel, brought me into this area.</p><p> </p><p><strong>What are you most excited about in terms of your move to Caltech?</strong></p><p>I think this is, of course, a fantastic place. The undergraduates here are very strong, and the graduate school is also very good, so I'm also very excited to work with all of those young people. Also, the physics department here is very good, and as I said, quantum field theory has recently provided promising new ways to think about these old problems from number theory. Caltech professors Anton Kapustin and Sergei Gukov have played central roles in revealing these connections between physics and the Langlands problem.</p><p> </p><p><strong>Is there anything else that you're looking forward to about living in Pasadena?</strong></p><p>I'm from Sichuan [province in China], and one thing that I miss is the food. It's hot and spicy, and now it's also kind of popular in the U.S. And there are very good Szechwan restaurants in the San Gabriel Valley. Actually, maybe the best Szechwan food in the U.S. is right here.</p><p> </p><p><strong>Aside from your research and professional interests, do you have any other hobbies?</strong></p><p>Yes, I've been playing the game Go for more than 20 years. It's a board game that is kind of like chess. It's interesting, and it's very complicated. Many years ago, you'd play with a game set and one opponent, but now you can also play it online. And that's good for me because after moving from place to place, it's hard to consistently find someone to play with.</p></div></div></div>Fri, 05 Dec 2014 22:43:48 +0000jsconrad45000 at http://www.caltech.eduUsing Simulation and Optimization to Cut Wait Times for Voters
http://www.caltech.edu/news/using-simulation-and-optimization-cut-wait-times-voters-44345
<div class="field field-name-news-writer field-type-ds field-label-inline clearfix"><div class="field-label">News Writer: </div><div class="field-items"><div class="field-item even">Jessica Stoller-Conrad</div></div></div><div class="field field-name-field-images field-type-file field-label-hidden"><div class="field-items"><div class="field-item even"><div class="ds-1col file file-image file-image-jpeg view-mode-full_grid_9 clearfix ">
<img src="http://s3-us-west-1.amazonaws.com/www-prod-storage.cloud.caltech.edu/styles/article_photo/s3/_McKenna-Sean_3727-GROUP-03-COMBO-NEWS-WEB%5B1%5D.jpg?itok=P57dIJMh" alt="" /><div class="field field-name-field-caption field-type-text field-label-hidden"><div class="field-items"><div class="field-item even">By developing a tool to help better prepare polling places, Caltech sophomore Sean McKenna is hoping to minimize the amount of time we spend in line at the polls.</div></div></div><div class="field field-name-credit-sane-label field-type-ds field-label-hidden"><div class="field-items"><div class="field-item even">Credit: Lance Hayashida/Caltech Office of Strategic Communications</div></div></div></div></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>No one ever likes long lines. Waiting in line may be inconvenient at the coffee shop or the bank, but it's a serious matter at voting centers, where a long wait time can discourage voters—and can be seen as an impediment to democracy.</p><p>However, with millions of Americans showing up at the polls, can long lines really be avoided on Election Day? By developing a tool to help better prepare polling places, Caltech sophomore Sean McKenna is using his Summer Undergraduate Research Fellowship (SURF) project as an opportunity to address that problem.</p><p>Over the summer, McKenna, an applied and computational mathematics major who works with Professor of Political Science <a href="http://www.hss.caltech.edu/content/r-m-alvarez">Michael Alvarez</a>, has been building a mathematics-informed tool that will predict busy times in precincts on Election Day and allocate voting machines in response to those predictions. This information could help election administrators minimize wait times for millions of voters.</p><p>"My project is based on a report from the Presidential Commission on Election Administration, which asserted that no American should ever have to wait more than 30 minutes to vote," McKenna says. "And so we're trying to see if we can help reach that goal by allocating voting machines in a new way."</p><p>McKenna's work is part of the <a href="http://vote.caltech.edu/">Caltech/MIT Voting Technology Project</a> (VTP), which has been working on voting technology and election administration since the 2000 election. At a June workshop for the collaborative VTP project, which aims to improve the voting process through research, McKenna met with academics and election administrators who suggested how he might apply his background in mathematics to create a tool for voting administrators to use on the VTP's website.</p><p>The tool he is developing uses a branch of applied mathematics called queueing theory to quantify the formation of lines on Election Day. "Queueing theory assumes that arrivals to a system like a polling place have a random, memoryless pattern. Under this assumption, the fact that one person just showed up to the precinct doesn't tell us whether the next person will show up two seconds from now or two minutes from now," he says. "Furthermore, queueing theory predicts line lengths and wait times as long-term averages, which scientists might call a steady-state approximation."</p><p>Although queueing theory provided a good jumping off point, there were a few real-world problems that an analytical model on its own couldn't address, McKenna says. For example, voter arrival behavior is <em>not</em> completely random on Election Day; early morning and late afternoon spikes in arrivals are the norm. In addition, polls are usually only open for 12 or 13 hours, which is not considered to be enough time for steady-state queueing approximations to be applicable.</p><p>"These challenges led us to review the literature and determine that running a simulation with actual data from administrators, as opposed to attempting to adjust strictly analytical models, was the best way to represent what actually happens in an election," McKenna explained.</p><p>The goal of the research is to create a simulation of an entire jurisdiction, such as a county with multiple polling places. The simulation would estimate wait times on Election Day based on information election administrators enter about their jurisdiction into the web-based tool. Administrators would then receive a customized output prior to Election Day, suggesting how to allocate voting machines across the jurisdiction and detailing the anticipated crowds—information that could both predict the severity of long lines and prompt new strategies for allocating voting machines to preempt long waits.</p><p>Several other Caltech undergraduates in Alvarez's group also have been working on alternative ways to improve the voting process. Senior physics major Jacob Shenker has been developing a system for more secure and user-friendly postal voting, and recent graduates Eugene Vinitsky (BS '14, physics) and Jonathan Schor (BS '14, biology and chemistry) produced a prototype of a mobile phone app that could help voters determine if there is a long line at their polling place.</p><p>While these projects were completed separately, McKenna says there may be room for collaboration in the future. "One thing that we're hoping my tool will be able to do is to predict for administrators what times are going to be busiest, and we could also export this information to the app for voters," he says. "For example, the app could alert someone that their polling place is very likely to have long lines in the morning so they should try to go in the afternoon."</p><p>The technologies that McKenna and his student colleagues are developing could change the way that millions of Americans participate in democracy in the future—which would be an impressive accomplishment for a young student who has yet to experience the physical aspect of lining up to vote.</p><p>"So that's one kind of sticky situation about my working on this project: I've never actually been in to vote in person. I've only been able to vote once, and since I'm from Minnesota, it had to be absentee by mail," he says.</p></div></div></div><div class="field field-name-field-pr-links field-type-link-field field-label-above"><div class="field-label">Related Links: </div><div class="field-items"><div class="field-item even"><a href="http://www.caltech.edu/content/technology-has-improved-voting-procedures" class="pr-link">Technology Has Improved Voting Procedures</a></div></div></div>Mon, 03 Nov 2014 17:07:29 +0000jsconrad44345 at http://www.caltech.eduMarkovic Elected to Great Britain's Royal Society
http://www.caltech.edu/news/markovic-elected-great-britains-royal-society-42732
<div class="field field-name-news-writer field-type-ds field-label-inline clearfix"><div class="field-label">News Writer: </div><div class="field-items"><div class="field-item even">Kimm Fesenmaier</div></div></div><div class="field field-name-field-images field-type-file field-label-hidden"><div class="field-items"><div class="field-item even"><div class="ds-1col file file-image file-image-jpeg view-mode-full_grid_9 clearfix ">
<img src="http://s3-us-west-1.amazonaws.com/www-prod-storage.cloud.caltech.edu/styles/article_photo/s3/Markovic_Vlad_0090-NEWS-WEB.jpg?itok=r8GfPb8-" alt="" /></div></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>Vladimir Markovic, the John D. MacArthur Professor of Mathematics at Caltech, has been named a fellow of Great Britain's Royal Society. <a href="https://royalsociety.org/about-us/fellowship/new-fellows-2014/">He is one of 50 new fellows and 10 foreign members elected in 2014</a>. Markovic's election brings to seven the number of fellows and foreign members of the Royal Society currently on the Caltech faculty.</p><p>Membership in the Royal Society is bestowed each year on a small number of the world's scientists. The oldest scientific academy in existence, the Royal Society was established in 1660 under the patronage of King Charles II for the purpose of "improving natural knowledge," and it helped usher in the age of modern science. Today, the society seeks to promote science leaders who champion innovation for the benefit of humanity and the planet.</p><p>Markovic studies the shapes and structures of mathematical spaces called manifolds. A line is a one-dimensional manifold while a plane would be two-dimensional. In its citation for Markovic, the Royal Society wrote, "Markovic is a world leader in the area of quasiconformal homeomorphisms and low dimensional topology and geometry. He has solved many famous and difficult problems. With Jeremy Kahn, he proved William Thurston's key conjecture that every closed hyperbolic 3-manifold contains an almost geodesic immersed surface."</p><p>In 2004, Markovic received awards recognizing his early career achievements from the London Mathematical Society and the Leverhulme Trust. In 2012, he was awarded the Clay Research Award. Earlier this year, he was an invited speaker at the International Congress of Mathematics in Seoul, South Korea.</p><p>Born in Germany, Markovic earned a BSc and PhD from the University of Belgrade in Serbia in 1995 and 1998, respectively. Before <a href="http://www.caltech.edu/content/particles-and-pants">joining the Caltech faculty as a professor in 2011</a>, he was an assistant professor at the University of Minnesota, an associate professor at SUNY Stony Brook, and a professor at the University of Warwick. He was named Caltech's John D. MacArthur Professor of Mathematics in 2013.</p><p> Markovic is currently on leave, teaching at the University of Cambridge.</p></div></div></div><div class="field field-name-field-pr-links field-type-link-field field-label-above"><div class="field-label">Related Links: </div><div class="field-items"><div class="field-item even"><a href="http://www.caltech.edu/content/particles-and-pants" class="pr-link">Particles and Pants: New Faculty Join PMA</a></div></div></div>Thu, 01 May 2014 23:00:42 +0000kfesenma42732 at http://www.caltech.eduHyperbolic Homogeneous Polynomials, Oh My!
http://www.caltech.edu/news/hyperbolic-homogeneous-polynomials-oh-my-42576
<div class="field field-name-news-writer field-type-ds field-label-inline clearfix"><div class="field-label">News Writer: </div><div class="field-items"><div class="field-item even">Cynthia Eller</div></div></div><div class="field field-name-field-images field-type-file field-label-hidden"><div class="field-items"><div class="field-item even"><div class="ds-1col file file-image file-image-jpeg view-mode-full_grid_9 clearfix ">
<img src="http://s3-us-west-1.amazonaws.com/www-prod-storage.cloud.caltech.edu/styles/article_photo/s3/pic%20for%20ramakrishnan%20story.jpg?itok=RqfWZ5Vn" alt="" /><div class="field field-name-field-caption field-type-text field-label-hidden"><div class="field-items"><div class="field-item even">Hyperbolic homogeneous equations on the chalkboard in Professor Dinakar Ramakrishnan's office at Caltech.</div></div></div><div class="field field-name-credit-sane-label field-type-ds field-label-hidden"><div class="field-items"><div class="field-item even">Credit: Cynthia Eller</div></div></div></div></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>Cutting-edge mathematics today, at least to the uninitiated, often sounds as if it bears no relation to the arithmetic we all learned in grade school. What do topology and combinatorics and <em>n</em>-dimensional space have to do with addition, subtraction, multiplication, and division? Yet there remains within mathematics one vibrant field of study that makes constant reference to basic arithmetic: number theory. Number theory—the "queen of mathematics," according to the famous 19<sup>th</sup> century mathematician Carl Friedrich Gauss—takes integers as its starting point. Begin counting 1, 2, 3, and you enter the domain of number theory.</p><p>Number theorists are particularly interested in prime numbers (those integers that cannot be divided by any number other than itself and 1) and Diophantine equations. Diophantine equations are polynomial equations (those with two or more variables) in which the coefficients are all integers.</p><p>It is these equations that are the inspiration for a recent proof offered by Dinakar Ramakrishnan, Caltech's Taussky-Todd-Lonergan Professor of Mathematics and executive officer for mathematics, and his coauthor, Mladen Dimitrov, formerly an Olga Taussky and John Todd Instructor in Mathematics at Caltech and now professor of mathematics at the University of Lille in France. This proof involves homogeneous equations: equations in which all the terms have the same degree. For example, the polynomial <em>xy </em>+<em> z</em><sup>2</sup> has degree 2, and <em>x</em><sup>2</sup><em>yz </em>+<em> xy</em><sup>3</sup> has degree 4. If we take an equation like <em>xy </em>=<em> z<sup>2</sup></em>, one solution for (<em>x, y, z</em>) would be (1, 4, 2). Multiplying that solution by any rational number will give infinitely many rational solutions, but this is a trivial way to get solutions achieved simply by "scaling." These are not the type of answers Ramakrishnan and Dimitrov were searching for.</p><p>What Ramakrishnan and Dimitrov showed is that a specific collection of systems of homogeneous equations with six variables has only a finite number of rational solutions (up to scaling). Usually people look for integer solutions of Diophantine equations, but the first approach is to find solutions in rational numbers—those that can be expressed as a fraction of two integers.</p><p>Diophantus, after whom the Diophantine equations are named, is best known for his <em><a href="http://www.caltech.edu/content/archimedes-revival-pasadena">Arithmetica</a>, </em>which Ramakrishnan describes as "a collection of intriguing mathematical problems, some of them original to Diophantus, others an assemblage of earlier work, some of it possibly going back to the Babylonians." Diophantus lived in the city of Alexandria, in what is now Egypt, during the third century CE. What makes the <em>Arithmetica </em>unusual is that it continues to serve as the basis for some very interesting mathematics more than 1,700 years later.</p><p>Diophantus was interested primarily in positive integers. He was aware of the existence of rational numbers, since he knew integers could divide one another, but he seemed to regard negative numbers (which are also rational numbers and can be integers) as absurd and unreal. Present-day number theorists have no such discomfort with negative numbers, but they continue to be as fascinated by integers as Diophantus was. "Integers are very special," says Ramakrishnan. "They are kind of like musical notes on a clavier. If you change a note even slightly, you'll hear a dissonance. In a sense, integers can be thought of as the well-tempered states of mathematics. They are quite beautiful."</p><p>Diophantus was especially interested in integer solutions for homogeneous polynomial equations: those in which each term of the equation has the same degree (for example, <em>x</em><sup>7</sup> + <em>y</em><sup>7</sup> = <em>z</em><sup>7</sup> or <em>x</em><sup>2</sup><em>y</em><sup>3</sup><em>z</em> = <em>w</em><sup>6</sup>). The classic example of a homogeneous polynomial equation is the Pythagorean theorem—<em>x</em><sup>2</sup> + <em>y</em><sup>2</sup> = <em>z</em><sup>2</sup>—which defines the hypotenuse, <em>z</em>, the longest side of a right triangle, with respect to the perpendicular sides <em>x</em> and <em>y</em>. As early as 1600 BCE, the ancient Babylonians knew that there were many integer solutions to this equation (beginning with 3<sup>2</sup> + 4<sup>2</sup> = 5<sup>2</sup>), though it was Pythagoras, a Greek mathematician living in the sixth century BCE, who gave his name to the formula, and Euclid who two centuries later proved that this equation has an infinite number of positive integer solutions, known as "Pythagorean triples" (such as 3, 4, 5; 5, 12, 13; or 39, 80, 89).</p><p>In 1637, French mathematician Pierre de Fermat famously wrote in the margin of Diophantus's <em>Arithmetica</em> that he had a "truly marvelous proof" showing that although there were an infinite number of positive integer solutions for <em>x</em><sup>2</sup> + <em>y</em><sup>2</sup> = <em>z</em><sup>2</sup>, there were no positive integer solutions at all when the variables were raised to the power of three or higher (<em>x</em><sup>3</sup> + <em>y</em><sup>3</sup> = <em>z</em><sup>3</sup>; <em>x</em><sup>4</sup> + <em>y</em><sup>4</sup> = <em>z</em><sup>4</sup> ; . . . ; <em>x<sup>n</sup></em> + <em>y<sup>n</sup></em> = <em>z<sup>n</sup></em>). Fermat did not provide the actual proof; he claimed that the margin of Diophantus's book was too small to contain it. Fermat's conjecture (it was not yet a proof, though Fermat apparently believed he had one in his mind) remained unsolved until the early 1990s, when British mathematician Andrew Wiles created a complicated and unexpected proof that made use of previously unrelated mathematical principles.</p><p>In geometric terms, Fermat's conjecture and Wiles's proof, with their three variables, operate in three-dimensional space and can be described as points on a curve on the projective plane, drawn with <em>x</em>, <em>y</em>, <em>z</em> coordinates up to scaling. By moving to a greater number of variables, Ramakrishnan and Dimitrov are interested in identifying points on so-called hyperbolic surfaces. A hyperbolic surface is a negatively curved space, like a saddle—as opposed to a positively curved space like a sphere—in which the rules of Euclidean geometry no longer apply. A simple example of a hyperbolic surface is given by the simultaneous solution (where the values of the variables are held constant) of three equations: <em>x</em><sub>1</sub><sup>5</sup> + <em>y</em><sup>5</sup> = <em>z</em><sup>5</sup>; <em>x</em><sub>2</sub><sup>5</sup> + <em>w</em><sup>5</sup> = <em>z</em><sup>5</sup>; and <em>x</em><sub>3</sub><sup>5</sup> + <em>w</em><sup>5</sup> = <em>y</em><sup>5</sup>. In the 1980s, German mathematician Gerd Faltings did pioneering work on the mathematics of hyperbolic curves, work that inspired Ramakrishnan and Dimitrov.</p><p>Ramakrishnan and Dimitrov's recent finding considers rational-number solutions for several systems of homogeneous polynomial equations describing hyperbolic surfaces. One solution is to set all the variables to zero. This solution is considered trivial; but are there any nontrivial solutions?</p><p>There are at least a few nontrivial solutions that Ramakrishnan and Dimitrov use as examples. Their challenge was to determine if there are finitely many or infinitely many rational solutions. They demonstrated—in a proof-by-contradiction that took nearly two years to complete—that the hyperbolic case they consider has only a finite number of solutions.</p><p>But, as Ramakrishnan remarks, there is no rest for number theorists, because "even if we solve another bunch of equations, there are still many more that are unsolved, enough for our descendants five hundred years from now."</p><p>For Ramakrishnan, this is not a counsel of despair. He continues to find mathematics exciting, especially the concept of the mathematical proof. As he points out, "In other ancient civilizations in the Middle East or India or China, they did some very complicated math, but it was more algorithmic, more related to computer science in my opinion than to philosophy. Whereas the Greeks emphasized proofs, rigorously establishing mathematical truths. There's nothing vague about it."</p><p>Apart from the inherent joy of pushing number theory forward through another generation, Ramakrishnan points out that this field has interesting applications in both science and everyday life. "Quite often in science, you are counting. Think of balancing chemical equations such as wCH<sub>4</sub> + xO<sub>2</sub> —> yCO<sub>2</sub> + zH<sub>2</sub>O, in which methane oxidizes to produce carbon dioxide and water. This is a linear Diophantine equation."</p><p>Number theory also plays an important role in encryption. "Every time one visits a website with an https:// address," says Ramakrishnan, "it is likely that the website browser is using an encryption system that validates the certificate for the remote server to which one is trying to connect. The security keys that are exchanged point to a number-theoretic solution. Most people prefer equations with simple solutions, but in some situations, such as encryption, you actually want integer equations that are hard to solve without the key. This is where number theory comes in."</p><p>Ramakrishnan and Dimitrov's paper, <a href="http://arxiv.org/abs/1401.1628">"Compact arithmetic quotients of the complex 2-ball and a conjecture of Lang,"</a> is posted on the math arXiv, a Cornell University Library open e-print archive for papers in physics, mathematics, computer science, quantitative biology, and quantitative finance and statistics.</p></div></div></div>Thu, 17 Apr 2014 16:33:10 +0000celler42576 at http://www.caltech.edu