IMA Related News
As our readers are well aware, the Institute for Mathematics and its Applications (IMA) is funded by the National Science Foundation and the University of Minnesota, and is closely associated with our department. September 2004 saw some changes in the leadership positions at the IMA. Professor Arnd Scheel is the new Deputy Director, replacing Professor Scot Adams who has taken on a no less challenging task as Director of Graduate Studies. Professor Fadil Santosa has stepped down as Deputy Director of the IMA, a position he held for seven years. He continues in his position as the Director of the MCIM (Minnesota Center for Industrial and Applied Mathematics). The new Associate Director is Debra Lewis, Professor of Mathematics at UC Santa Cruz, who has taken a two-year leave from UC Santa Cruz to serve in this position and as an Adjunct Professor in the School of Mathematics.
Professor Douglas Arnold continues to serve as the Institute’s Director. He wrote eloquently: “I am delighted to be working with Arnd and Debra. Their commitment and skill are already much in evidence here. I am also very grateful to Fadil and Scot for all they did in their time at the IMA. They made major contributions to the quality and reputation of the IMA, and are both continuing to serve with major roles in the department. Fadil deserves a special thanks for his exceptional commitment, having worked at the IMA not just since I came here, but also with the two previous directors. As one of the leading mathematicians worldwide in the fostering of the application of mathematical research in industry, he has built and developed IMA’s industrial programs, and has also been invaluable in almost every other aspect of the IMA: scientific program development, postdoctoral recruiting and mentoring, publications, publicity, assessment, administration, etc.”
Professor Scheel’s field of research is dynamical systems, especially the dynamics of PDE and the formation and stability of patterns. He has been involved in many IMA programs in the past, beginning with a month he spent here in 1997, before joining the department in 2001.
Professor Lewis’ research is in the area of geometric mechanics, particularly Hamiltonian and Lagrangian systems with symmetry. She received her PhD in Berkeley in 1987 and spent 1988-1989 in Minnesota as an IMA postdoc.
Each year, the work of IMA researchers focuses on one major area of applications of mathematics, with the 2004-2005 focus being on “Mathematics of Materials and Macromolecules: Multiple Scales, Disorder, and Singularities”. Professor Mitchell Luskin serves as the Chair of the organizing committee which also includes Professor Carme Calderer. In addition, as is the case each year, several department members with related specialities also participate in the program and in mentoring the IMA postdoctoral researchers. This year these include Professors Maury Bramson, Robert Gulliver, George Sell and Vladimir Sverak. Professor Luskin was kind enough to give us the following note describing the 2004-2005 program.
IMA Year on Mathematics of Materials and Macromolecules:
Multiple Scales, Disorder, and Singularities
by Mitchell Luskin
During the academic year 2004-2005, the IMA is hosting a program aimed at a synthesis of the problems at the interface between mathematics, materials science, condensed matter physics, and biology. The program organizers (including Mitchell Luskin, Chair, and Carme Calderer) have developed a program to provide rich interdisciplinary opportunities for the interplay between mathematics and emerging approaches to the study of matter and macromolecules. We are particularly focusing on phenomena that require modeling that integrates the atomic to the continuum scales.
IMA researchers are utilizing a broad spectrum of modern mathematics to understand matter. Several of the most active research efforts use nonlinear partial differential equations to model the structure and dynamics of defects and microstructure. Current research in stochastic differential equations is being utilized to model microscale and nanoscale devices and phenomena, and topological and geometric concepts are being developed to understand defects in crystals, the structure of DNA, and protein folding.
IMA mathematicians and materials scientists are confronting the computational challenges of multiscale modeling, singularities, and disorder. Contemporary computational algorithms for the study of matter, such as “hyperdynamics,” are often developed in the context and language of physical theories that are not part of the traditional education of computational mathematicians. This IMA program is striving to enable interactions and research between mathematicians and materials scientists on such computational problems that are important in the study of matter, but have been given little attention by the mathematics community. For more details about the annual program please consult the IMA website http://www.ima.umn.edu/
A glance at the 2004 and 2005 Summer Programs also helps to illustrate the great scope and timeliness of investigations being pursued by IMA researchers. The Program on n-Categories (June 7-18, 2004), organized by Peter May (U. of Chicago) and John Baez (UC Riverside), was of great interest to members of our department, with participants including Professors Scot Adams, Bernard Badzioch, Mark Feshbach, Mehdi Hakim Hashemi, William Messing, Victor Reiner, Fadil Santosa, Sasha Voronov, and Peter Webb, as well as graduate students Eric Harrelson, Hyeung-Joon Kim, Jonathan Rogness, James Swenson, and Javier Zuniga. As is well known to many of our readers, the category theory introduced in 1945 by S. Eilenberg and S. MacLane provided a powerful new language for mathematics. Many of the leading researchers are now working on an even more potent “higher category theory promis[ing] to allow... study of higher categorical structures that appear in a variety of specific fields. The need for such a language has become apparent, almost simultaneously, in mathematical physics, algebraic geometry, computer science, logic”.
The Principal Speakers at the Program on Computational Topology (July 6-16, 2004) were H. Edelsbrunner and J. Harer, both of Duke University. Computational Topology “grew out of Computational Geometry as researchers expanded into applications where significant topological issues arise. The two such areas discussed [were] structural molecular biology and geometric modeling. Both have connections to industries of substantial economical size.”
The Wireless Communications Program taking place June 22-July 1, 2005 will encompass short courses and research presentations in the following relevant areas of pure and applied mathematics: stochastic calculus, information theory, signal processing, optimization, and control theory. As always, the objective is to facilitate interaction between academia and industry, mathematicians and engineers.
The meeting “New Directions in Probability Theory” (August 5-6, 2005) will involve major participation by our faculty. Professors Maury Bramson and Ofer Zeitouni are among the organizers and Professor Zeitouni is also one of the main speakers. In addition, the organizers include Professor Mike Cranston (UC Irvine), a 1980 graduate of our department. The topics to be covered include Flows and Random Media; Probability, Combinatorics, and Statistical Mechanics; Stochastic Integration; Stochastic Partial Differential Equations; and Random Walk in Random Environment.
A Workshop for Graduate Students on Mathematical Modeling in Industry will take place August 1-10, 2005. The New Directions Short Course on Quantum Computation (August 15-26, 2005) will be taught by Peter W. Shor (MIT) and Alexei Kitaev (Caltech). Professor Shor is one of the founders of this new area which may have the potential to bring about yet another revolution in computing.
IMA PUBLIC LECTURES
These lectures, given by world’s leading experts in their areas, attract large audiences of the general public. The following reports on the lectures by Sir Roger Penrose and Professor James Murray were provided by Todd Wittman, a grad student in the School of Mathematics. Todd was introduced to our readers in last year’s newsletter as recipient of several awards for teaching excellence. We thank him for these lively reports.
Most scientists and certainly all mathematicians would answer the title of Sir Roger’s talk with a resounding “yes”. But there appears to be an inherent contradiction in that assertion: How could mathematics rule the world when the physical world existed long before mathematics was spawned by human imagination? Are we discovering the physical laws that describe the physical world or are we creating these laws in an attempt to order our universe? And if we are forcing a mathematical square peg into the universe’s round hole, what hope is there for the future of physical research? These questions seem a bit too philosophical for a math talk, but Sir Roger gave his opinions on the subject through a discussion of quantum physics.
Sir Roger is a world-renowned mathematician who has made significant advances in mathematical physics, specifically quantum theory. He has written several important papers in topology and introduced twistor theory to connect relativity to quantum theory. He has authored several popular books, including “The Emperor’s New Mind” and “The Nature of Space and Time”, co-authored with the physicist Stephen Hawking. Given his credentials, it may seem a bit odd for Sir Roger to speak to a standing-room only crowd that included undergraduates, high school students, families, and other non-mathematicians. The ability to communicate mathematical research to the general public is a very rare skill, and this has certainly contributed to the poor reputation of mathematics among the general public. Sir Roger rose to the occasion, using colorful illustrations he prepared on transparencies, analogies to explain complex physical phenomena, and a friendly and lively speaking tone.
Sir Roger explained some basic concepts through analogies, such as the famous Schrodinger’s cat example to describe the superposition of quantum states. If we accept the world of classical physics as being clear as the light of day, then the probabilistic world of quantum physics can be compared to the murky depths beneath the sea. Emphasizing the importance of mathematics as a bridge between classical and quantum physics, Sir Roger suggested mathematics is a mermaid, able to breathe and thrive in both worlds equally well.
To solve the paradox between reality and our attempt to describe it, Sir Roger proposed the existence of three worlds: the physical, mental, and platonic. The physical world, having given rise to humans, gave rise to the mental world. The mental world in turn created the platonic world of mathematics and physical theory. The platonic world represents our best efforts to describe the physical world in which we live. If we can accept the existence of a “real” world, then we should be able to accept that advances in mathematics and rational thought will allow us to discover that world. Those discoveries may be filtered through the mental world of the human mind, but that is just an unavoidable consequence of living on this world rather than in it.
James D. Murray, University of Oxford & University of Washington, November 18, 2004: “The Marriage Equation: A Practical Theory for Predicting Divorces and a Scientifically-Based Marital Therapy”
Prof. James Murray from the University of Washington is a mathematical biologist who holds the distinction of doing research in just about every area of mathematical biology, from epidemiology to medicine to ecology. He has been involved in many interesting and important research projects, ranging from modeling the growth of brain tumors to describing how the leopard got its spots. A few years ago, Prof. Murray engaged in his most radical and perhaps most challenging research project to date: making marriages work.
Prof. Murray was approached by a psychology professor at the University of Washington working in marital therapy. The psychologist and his team of graduate students had videotaped over 800 married couples in marital therapy sessions. The team then reviewed the tape and literally charted the course of the conversations. With a very well-defined scale taking into account tone and facial expressions, the researchers assigned a value from -10 to +10 assessing the “negativity” or “positiveness” of the response. This gave the psychologists a huge set of time series data to work with, but they weren’t sure how to work with it. So they turned to a mathematician.
Examining the data, Murray viewed each person’s response as a dynamical system. He modeled the response as a linear recurrence relation, depending on an individual's last response as well as their partner’s last response. This gave a mathematical proof of the existence of stable nodes, literally sinks into which the conversation would get stuck. The goal of a marital therapist is to ensure that the session ends positively. Mathematically speaking, if we make a 2D scatterplot of the conversation with the individual’s responses as the two axes, the marital therapist should try to ensure that the stable node lies in the first quadrant. This gave rise to a new approach to marriage counseling as well as a new area of mathematical research. Prof. Murray published, with his colleagues, the book “The Mathematics of Marriage: Dynamic Nonlinear Models.”
The conclusion of this study was that couples that work best include people with matching personalities. That is, two passive individuals will be more likely to form a stable marriage than one passive and one volatile individual. Oddly enough, two volatile individuals can form a stable marriage, although Murray warned that the relationship may consist of “alternating cycles of fighting and sex.”
Speaking to a large, mostly non-mathematical audience, Prof. Murray outlined his so-called “Marriage Equation” in simple and easy-to-grasp terms. Aware that this was a public lecture, the highest level mathematics used was the equation of a line, yet Prof. Murray still conveyed deep insight into mathematical research in the social sciences. With his lilting Scottish accent and jovial speaking style, Murray kept the audience’s interest with humorous anecdotes and mathematical insight into the human heart. To convince his audience of the accuracy of his mathematical model, his research team did a follow-up study on the 800 couples first used to gather the conversation data. With frightening accuracy, they were able to mathematically predict which couples would get divorced and which would stay together. So mathematics may not be able to make people happy, but it can predict which ones will be happy.
We recently had two additional IMA Public Lectures which will be reported in detail in next year’s newsletter. The titles and speakers were “Math Behind the Curtains: Dynamic Simulation at Pixar” (February 9, 2005), by David Baraff, Senior Animation Scientist, Pixar Animation Studios, and “Computers and the Future of Mathematical Proof” (March 30, 2005), by Thomas C. Hales, Mellon Professor of Mathematics, University of Pittsburgh.