Caltech News tagged with "mathematics"
http://www.caltech.edu/news/tag_ids/40/rss.xml
enPrime 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=_gIz95SI" 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 Marketing and 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=LAYfLX8e" 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 Marketing and 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=1RTJOmOC" 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=zDL13dNc" 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.eduA Mathematical Approach to Physical Problems: An Interview with Rupert Frank
http://www.caltech.edu/news/mathematical-approach-physical-problems-interview-rupert-frank-41206
<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/Frank-R_3715-NEWS-WEB.jpg?itok=kBF7pP4G" alt="" /><div class="field field-name-field-caption field-type-text field-label-hidden"><div class="field-items"><div class="field-item even">Rupert Frank, 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 Marketing and 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><a href="http://www.math.caltech.edu/~rlfrank/">Rupert Frank</a> joined the Caltech faculty this spring as a professor of mathematics. Originally from Munich, Germany, Frank graduated from the Ludwig Maximilian University in his hometown in 2003 and his PhD from the Royal Institute of Technology in Stockholm, Sweden, in 2007. After completing a postdoctoral position at Princeton University, he was hired as an instructor there and quickly worked his way up to assistant professor. Frank recently answered a few questions about his work at the intersection of mathematics and physics.</em></p><p><strong>What do you work on?</strong></p><p>I work in this area called mathematical physics. It involves taking things that we see and observe in nature and trying to explain them mathematically from first principles. In mathematics, people often say that they're doing algebra or geometry or something, where they are talking about the methods they are using. However, for us it's more that we use whatever methods we need in order to understand a concrete problem. It's much more problem-specific.</p><p>For example, one thing that we still cannot explain—that we are actually really far from being able to explain—is the emergence of periodic structures; that is, structures that repeat themselves. It's clear in nature that it does happen. We see crystals, for example. But we still have no idea why this happens. It's embarrassing really.</p><p><strong>So how do you approach a problem like that?</strong></p><p>We like to start, for example, with the rules of quantum mechanics—some axioms, which describe the state and the energy of a system. From there, we would like to see that periodic structures can emerge on a macroscopic scale.</p><p>Sometimes we work with smaller dimensions—one-dimensional or two-dimensional models, not three dimensional, as nature is. Or we work with discrete models where you assume that all objects can only sit at discrete sites; they cannot move continuously through space. There is a hope that by working with such models, one can reveal more about the overall system.</p><p><strong>What problems are you currently addressing?</strong></p><p>An important aspect of my work is symmetry and symmetry breaking. Periodicity is a particular case of symmetry.</p><p>A problem that I'm always working on is how to explain superconductivity. Superconductivity is a quantum phenomenon that happens on a macroscopic scale, meaning that I can observe it with my bare eyes. [The phenomenon involves the electrical resistance of certain metals and ceramics dropping to zero when cooled below a particular critical temperature. This means such materials can conduct electricity for longer periods, more efficiently. They also repel magnetic fields.] But I cannot explain it with ordinary classical mechanics; I need quantum mechanics. So again, the point is how do we come up with a theory for superconductivity on a macroscopic scale from a microscopic model using the laws of quantum mechanics? And that has been understood, I would say, on a physical level, and there are models that work numerically very well, but mathematically it has not been clarified.</p><p><strong>How would you say the discipline of mathematical physics informs both mathematics and physics?</strong></p><p>Well, mathematics and physics have always been interrelated, and a lot of mathematics has been developed while trying to solve physical problems. I think physics, from a mathematics perspective, leads to interesting mathematical problems. You are trying to prove something, and it's typically related to some optimization problem—where you want to minimize energy costs or something. So it gives you a way of thinking.</p><p>In terms of the benefit to physics, I think we can sometimes provide a different perspective. Physicists typically speak about what they consider to be typical cases within a model, whereas in mathematics, one usually works on the negative side—trying to exclude the atypical. So from time to time, we come up with problems that really require physical explanation that has not been there before.</p><p><strong>How did you originally become interested in mathematics and physics?</strong></p><p>Actually, both my mother and my father are mathematicians, and one of my brothers is a mathematician; the other is a computer scientist. So it was around when I was growing up, that's for sure. By my third year of university studies, I knew which field of mathematics I wanted to focus on. It can be called functional analysis, operator theory, or mathematical physics. And I saw that all of this was intrinsically related to quantum mechanics. To a certain extent, this field of mathematics was created to explain quantum mechanics. So it was clear that I had to go into physics.</p><p><strong>Why did you decide to come to Caltech?</strong></p><p>Well, it's a very nice place, and it's a smaller place. That gives you a lot of opportunities because you're not only one of the many. Everybody expects you to do something, and they help you to do it. That's something that I really appreciate.</p></div></div></div>Mon, 18 Nov 2013 22:34:37 +0000ksvitil41206 at http://www.caltech.eduReducing Coincidence with Mathematics: An Interview with Nets Katz
http://www.caltech.edu/news/reducing-coincidence-mathematics-interview-nets-katz-40725
<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/NKatz_7578-NEWS-WEB%5B1%5D.jpg?itok=GxJEnJ92" 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><em>Raised in Grand Prairie, Texas, Nets Katz began pursuing mathematics at an early age, earning a bachelor's from Rice University in 1990 at the age of 17 and a doctorate from the University of Pennsylvania in 1993 at 20. After completing several postdoctoral fellowships, Katz went on to an assistant professorship at the University of Illinois at Chicago, an associate professorship at Washington University in Saint Louis, and a full professorship at Indiana University before joining the faculty at Caltech in January. Recently, Katz answered a few questions about his move to Southern California and his research in a field of math called combinatorics.</em></p><p><strong>What brought you to Caltech, and why are you excited to be here?</strong></p><p>I was offered a job here! It's a great institution; I've always admired it. Tom Wolff, who was a math professor here about 12 years ago, was a major influence in my career, and I flatter myself to think that I'm continuing some of his work.</p><p>I'm also really excited about teaching these students. I'm teaching Math 1 in the fall, and I'm really looking forward to the unique opportunity to get across deep and useful ideas to the very best students, including students who aren't in my field. Mathematics has always had a significant impact on the other sciences and engineering, and I think it will continue to do so.</p><p><strong>What are your research interests?</strong></p><p>I'm interested in showing that you can't have very many coincidences. The problems that I'm interested in are mostly considered to be in combinatorics [a field of math concerned with finding maximum, minimum, and optimum configurations—such as the absolute largest or smallest possible size of an object]. In a problem I worked on a few years ago, called the Erdos distance problem, we wanted to know the minimum number of distinct distances possible between a set number of points in a plane.</p><p>For example, say you're playing a game in which you ask an opponent to draw a finite number of <em>n</em> dots on a sheet of paper; the object of the game is for the opponent to position their dots so that the number of distinct distances between the dots is as small as possible. You then determine how well they did by tallying the number of distinct distances between these dots. If the dots are truly positioned randomly, some of the distances between the dots <em>could</em> be the same, but almost all of the distances will be different—meaning your opponent didn't do very well.</p><p>Placing the dots in a grid-like lattice pattern would be a relatively good strategy to win the game—this arrangement allows you to position the dots in such a way that a lot of the distances are the same. When working on this problem, we were able to prove that you can't get fewer than <em>n</em>/log <em>n</em> distinct distances—which, surprisingly, means that there isn't a strategy much better than the lattice. If there were a better strategy or design, it would involve a lot of coincidences—and too many coincidences aren't possible. You have to have some really incredibly special design to come close to the lattice arrangement, and what we were able to show is that even the best "incredibly special design" really isn't better than the lattice by very much.</p><p><strong>How did you first become interested in math?</strong></p><p>My father was a physicist, so we would have conversations about math at the dinner table. Of course, he was very far from a mathematician, but among physicists in his day he was quite well versed in math. And he had criticisms of how math was done, not all of which made sense, so we would have discussions about the foundations of things that were really exciting.</p><p>Here we were in Grand Prairie—I, a little kid no one had ever heard of, and he, a physicist largely forgotten—talking about how we might set the foundations of mathematics in a more clever way than all the denizens of the MITs and Caltechs of the world had ever managed to do. It was incredibly empowering. If it weren't constrained by the substantive requirements of mathematics, it might have been megalomaniacal. He made me feel that a person out of nowhere could really change the way people think about things. This was very exciting to me.</p><p><strong>How do you like living in Southern California?</strong></p><p>Actually, I have to admit I'm much more of an Indiana person—I prefer small towns and rural areas to densely populated cities—so I find I am experiencing a lot of culture shock living in Southern California. But there are nice things about the location; I have a lot of friends at UCLA and being closer is definitely a plus.</p></div></div></div>Thu, 17 Oct 2013 17:54:22 +0000jsconrad40725 at http://www.caltech.eduNotes from the Back Row: "Quantum Entanglement and Quantum Computing"
http://www.caltech.edu/news/notes-back-row-quantum-entanglement-and-quantum-computing-39378
<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">Douglas Smith</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/Watson_Lecture-Preskill-NEWS-WEB.jpg.jpeg?itok=jIDiTjjc" 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><span style="line-height: 1.538em;">John Preskill, the Richard P. Feynman Professor of Theoretical Physics, is hooked on quanta. He was applying quantum theory to black holes back in 1994 when mathematician Peter Shor (BS '81), then at Bell Labs, showed that a quantum computer could factor a very large number in a very short time. Much of the world's confidential information is protected by codes whose security depends on numerical "keys" large enough to not be factorable in the lifetime of your average evildoer, so, Preskill says, "When I heard about this, I was awestruck." The longest number ever factored by a real computer had 193 digits, and it took "several months for a network of hundreds of workstations collaborating over the Internet," Preskill continues. "If we wanted to factor a 500-digit number instead, it would take longer than the age of the universe." And yet, a quantum computer running at the same processor speed could polish off 193 digits in one-tenth of a second, he says. Factoring a 500-digit number would take all of two seconds.</span></p><p>While an ordinary computer chews through a calculation one bite at a time, a quantum computer arrives at its answer almost instantaneously because it essentially swallows the problem whole. It can do so because quantum information is "entangled," a state of being that is fundamental to the quantum world and completely foreign to ours. In the world we're used to, the two socks in a pair are always the same color. It doesn't matter who looks at them, where they are, or how they're looked at. There's no such independent reality in the quantum world, where the act of opening one of a matched pair of quantum boxes determines the contents of the other one—even if the two boxes are at opposite ends of the universe—but <em>only</em> if the other box is opened in exactly the same way. "Quantum boxes are not like soxes," Preskill says. (If entanglement sounds like a load of hooey to you, you're not alone. Preskill notes that Albert Einstein famously derided it back in the 1930s. "He called it 'spooky action at a distance,' and that sounds even more derisive when you say it in German—'<em>Spukhafte Fernwirkungen!</em>'")</p><p>An ordinary computer processes "bits," which are units of information encoded in batches of electrons, patches of magnetic field, or some other physical form. The "qubits" of a quantum computer are encoded by their entanglement, and these entanglements come with a big Do Not Disturb sign. Because the informational content of a quantum "box" is unknown until you open it and look inside, qubits exist only in secret, making them ideal for spies and high finance. However, this impenetrable security is also the quantum computer's downfall. Such a machine would be morbidly sensitive—the slightest encroachment from the outside world would demolish the entanglement and crash the system.</p><p>Ordinary computers cope with errors by storing information in triplicate. If one copy of a bit gets corrupted, it will no longer match the other two; error-detecting software constantly checks the three copies against one another and returns the flipped bit to its original state. Fixing flipped bits when you're not allowed to look at them seems an impossible challenge on the face of it, but after reading Shor's paper Preskill decided to give it a shot. Over the next few years, he and his grad student Daniel Gottesman (PhD '97) worked on quantum error correction, eventually arriving at a mathematical procedure by which indirectly measuring the states of five qubits would allow an error in any one of them to be fixed.</p><p>This changed the barriers facing practical quantum computation from insurmountable to merely incredibly difficult. The first working quantum computers, built in several labs in the early 2000s, were based on lasers interacting with what Preskill describes as "a handful" of trapped ions to perform "a modest number of [logic] operations." An ion trap is about the size of a thermos bottle, but the laser systems and their associated electronics take up several hundred square feet of lab space. With several million logic gates on a typical computer chip, scaling up this technology is a <em>really</em> big problem. Is there a better way? Perhaps. According Preskill, his colleagues at Caltech's Institute for Quantum Information and Matter are working out the details of a "potentially transformative" approach that would allow quantum computers to be made using the same silicon-based technologies as ordinary ones.</p><p><strong> "</strong><a href="https://itunes.apple.com/us/podcast/quantum-entanglement-quantum/id438265074?i=147270494&mt=2"><strong>Quantum Entanglement and Quantum Computing</strong></a><strong>" is available for download in HD from Caltech on iTunesU. (Episode 19)</strong></p></div></div></div>Thu, 09 May 2013 17:39:14 +0000dsmith39378 at http://www.caltech.eduOscar Bruno Named SIAM Fellow
http://www.caltech.edu/news/oscar-bruno-named-siam-fellow-39271
<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">Brian Bell</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/OscarBruno_0.jpg?itok=1YD_s2sB" 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>The Society for Industrial and Applied Mathematics (SIAM) named Oscar P. Bruno, professor of applied and computational mathematics at Caltech, as a member of its 2013 Class of Fellows.</p><p>Bruno is one of 33 fellows selected by SIAM for "exemplary research as well as outstanding service to the community," according to the organization. "Through their contributions, the 2013 Class of Fellows is helping advance the fields of applied mathematics and computational science," the organization stated in a March 29 press release.</p><p>"SIAM is an organization that includes many of the leading applied mathematicians from around the world, so I am honored to have been selected for their 2013 Class of Fellows," Bruno says.</p><p>Bruno's research group at Caltech focuses on the development of accurate high-performance numerical partial differential equation solvers applicable to realistic scientific and engineering configurations. His research interests include numerical analysis, multiphysics modeling and simulation, and mathematical physics.</p><p>Bruno graduated with a Friedrichs Prize for an outstanding dissertation in mathematics from the Courant Institute of Mathematical Sciences at New York University in 1989. He became an associate professor at Caltech in 1995 and a professor of applied and computational mathematics in 1998. He has served as executive officer of Caltech's Applied and Computational Mathematics department, and he is the recipient of a Young Investigator Award from the National Science Foundation and a Sloan Foundation Fellowship.</p><p>Bruno is a member of the SIAM Council, and he serves on the editorial boards of the <em>SIAM Journal on Scientific Computing</em> and the <em>SIAM Journal on Applied Mathematics</em>.</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.cms.caltech.edu/" class="pr-link">Oscar Bruno's page</a></div><div class="field-item odd"><a href="http://www.siam.org" class="pr-link">Society for Industrial and Applied Mathematics website</a></div><div class="field-item even"><a href="http://connect.siam.org/siam-announces-class-of-2013-fellows/" class="pr-link">SIAM Fellowship press release</a></div></div></div>Mon, 22 Apr 2013 23:13:33 +0000bbell239271 at http://www.caltech.eduQuantum Entanglement and Quantum Computing
http://www.caltech.edu/news/quantum-entanglement-and-quantum-computing-39090
<div class="field field-name-field-subtitle field-type-text field-label-hidden"><div class="field-items"><div class="field-item even">Watson Lecture Preview</div></div></div><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">Douglas Smith</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/Watson_Lecture-Preskill-NEWS-WEB.jpg?itok=HYYrUBtm" 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>John Preskill, the Richard P. Feynman Professor of Theoretical Physics, is himself deeply entangled in the quantum world. Different rules apply there, and objects that obey them are now being made in our world, as he explains at <a href="/content/john-preskill-quantum-entanglement-and-quantum-computing">8:00 p.m. on Wednesday, April 3, 2013</a>, in Caltech's Beckman Auditorium. Admission is free.</p><p> </p><p><strong>Q: What do you do?</strong></p><p>A: I'm trying to understand what a quantum computer would be capable of, how we could build one, and whether it would really work. My background is in particle theory, a subject I still love, but in the spring of 1994 a mathematician at Bell Labs named Peter Shor [BS 1981] discovered an algorithm for factoring large numbers with a quantum computer. I got really excited by this, because it moved the boundary separating "easy" problems, which we can eventually expect to solve with advanced technologies, from truly hard problems that we may never be able to solve. There are problems we can solve using quantum physics that we couldn't solve otherwise. The crucial problem is protecting a quantum computer from the various kinds of "noise" that could destroy quantum entanglement, and we've made a lot of progress on that.</p><p> </p><p><strong>Q: OK, so what's "entanglement?"</strong></p><p>A: It's the correlations between the parts of a system. Suppose you have a 100-page book with print on every page. If you read 10 pages, you'll know 10 percent of the contents. And if you read another 10 pages, you'll learn another 10 percent. But in a highly entangled quantum book, if you read the pages one at a time—or even 10 at a time—you'll learn almost nothing. The information isn't written on the pages. It's stored in the correlations among the pages, so you have to somehow read all of them at once.</p><p>There's another important difference: If Alice and Bob both read this morning's <em>New York Times</em>, they will have perfectly correlated information. And if Charlie comes along and reads the same paper later on, he will be just as strongly correlated with Alice as Alice is with Bob, and Bob will be just as correlated with Charlie as he is with Alice. But if Alice reads her quantum newspaper and Bob reads his, they will learn almost nothing until they get together and share their information. Now, when Charlie comes along, Alice and Bob have already used up all their ability to be entangled, and he's completely left out. Entanglement is monogamous—if Alice and Bob are as entangled as they can be, neither of them can entangle with Charlie at all. So if Alice wants to be entangled with <em>both</em> Bob and Charlie, there's a limit to how entangled she can be with either one. They have to work out some sort of compromise.</p><p> </p><p><strong>Q: What gets you excited about this?</strong></p><p>A: The technology is emerging to make it possible to do things we've never done before. We were taught in school that classical physics applies to things you can see, and quantum physics applies to the world at the scale of atoms and below. We're rebelling against that by making systems that are big enough to see, yet still exhibit quantum behavior. For example, Professor of Applied Physics Oskar Painter [MS 1995, PhD 2001] has made a tiny silicon bar that's suspended in space, and he's successfully cooled it all the way down to its quantum-mechanical ground state. It vibrates in a mode that corresponds to its lowest quantum state. He hasn't entangled such bars yet, but he knows how to do it.</p><p>We're exploring a new frontier of physics. It's not the frontier of short distances, like in particle physics; or of long distances, like in cosmology. It's what you might call the entanglement frontier.</p><p> </p><p><strong><em>Named for the late Caltech professor Earnest C. Watson, who founded the series in 1922, the Watson Lectures present Caltech and JPL researchers describing their work to the public. Many past Watson Lectures are available online at </em></strong><a href="http://itunes.apple.com/us/itunes-u/watson-lectures-sd/id422627541"><strong><em>Caltech's iTunes U</em></strong></a><strong><em> site.</em></strong></p></div></div></div>Tue, 02 Apr 2013 16:35:30 +0000dsmith39090 at http://www.caltech.eduCaltech Faculty Named AMS Fellows
http://www.caltech.edu/news/caltech-faculty-named-ams-fellows-37310
<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">Marcus Woo</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/AMS_news2b_0.jpg?itok=kFzXS6eB" 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>Nine members of Caltech's faculty have been named to the inaugural class of Fellows of the American Mathematical Society (AMS). According to the AMS, the distinction recognizes members of the society "who have made outstanding contributions to the creation, exposition, advancement, communication, and utilization of mathematics." There are 1,119 Fellows in the inaugural class from more than 600 institutions.</p><p>Caltech's Fellows are</p><p><strong>Tom Apostol</strong>, Professor of Mathematics, Emeritus</p><p><strong>Michael Aschbacher</strong>, Shaler Arthur Hanisch Professor of Mathematics</p><p><strong>Danny Calegari</strong><em>,</em> Richard Merkin Distinguished Professor of Mathematics</p><p><strong>Yizhao Thomas Hou</strong>, Charles Lee Powell Professor of Applied and Computational Mathematics</p><p><strong>Alexander Kechris</strong>, Professor of Mathematics</p><p><strong>Wilhelmus Luxemburg</strong>, Professor of Mathematics, Emeritus</p><p><strong>Hirosi Ooguri</strong>, Fred Kavli Professor of Theoretical Physics and Mathematics</p><p><strong>Dinakar Ramakrishnan</strong>, Taussky-Todd-Lonergan Professor of Mathematics</p><p><strong>Barry Simon</strong>, International Business Machines Professor of Mathematics and Theoretical Physics</p><p>"I'm delighted that the American Mathematical Society has recognized so many of our mathematics faculty for inclusion in their first class of Fellows," says Tom Soifer, chair of the Division of Physics, Mathematics and Astronomy. "This is a clear recognition of the quality and stature of our mathematics faculty by the world's largest and most influential society for mathematical research and education."</p></div></div></div>Thu, 01 Nov 2012 23:06:42 +0000mwoo37310 at http://www.caltech.edu