Colloquium Lectures
Special Undergraduate Lecture April 16
Wednesday April 16
Lecture: 3:30 in Damen 439
Refreshments: 3:00 in Math Department -- Damen 310
Speaker: Daniel Sternheimer, Keio University, Japan
Title: Deformations, Quantizations, and the Geometry of Space-Time:
An Introductory Overview
Abstract: We present, from an epistemological point of view, the evolution of physical concepts in the context of the relation between mathematics and physics. We stress the importance of symmetries and of space-time in fundamental physical theories and show that the above evolution is best understood in the framework of the mathematical notion of deformation. The concepts of relativity and of quantization are
important paradigms.
In the last part we explain how deforming the space-time of Einstein, Lorentz and Minkowski and its Lie group of symmetries leads to a fruitful object which together with its group of symmetries is referred as AdS or "anti de Sitter space". The study of AdS has significant physical consequences. One example is that massless
particles in four dimensional space-time like photons become, in a way compatible with quantum electrodynamics, composites of massless particles in three dimensional space-time called singletons. We end by describing an ongoing program in which anti de Sitter would be quantized in some regions related to black holes, speculating that this might explain the creation of matter in a universe in accelerated expansion.
About the Speaker: Dr. Sternheimer is one of the world's most renowned experts in the theory and use of deformations and quantizations in mathematical physics. In particular, his seminal papers "Deformation Theory and Quantization I, II" (Annals of Physics, 1978) paved the way for the interpretation of quantum mechanics as a deformation of classical mechanics. Dr. Sternheimer was a member of the Laboratoire de Physique Mathematique at the Universite de Bourgogne for 35 years. He is also editor of the distinguished journal Letters in Mathematical Physics for many years and he serves as editor of the Modern Encyclopedia of Mathematical Physics. According to MathSciNet, his publications have been cited 399 times by 267 authors.
Undergraduate Colloquium
Friday April 4
Lecture at 3:00 in Damen 517
Refreshments at 2:30 in Math Department -- Damen 310
Speaker: David Bressoud, Macalaster College
Title: Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture
Abstract: What is the role of proof in mathematics? Most of the time, the search for proof is less about establishing truth than it is about exploring unknown territory. In finding a route from what is known to the result one believes is out there, the mathematician often encounters unexpected insights into seemingly unrelated problems. I will illustrate this point with an example of recent research into a generalization of the permutation matrix known as the "alternating sign matrix." This is a story that began with Charles Dodgson (aka Lewis Carroll), matured at the Institute for Defense Analysis, drew in researchers from combinatorics, analysis, and algebra, and ultimately was solved with insights from statistical mechanics. This talk is intended for a general audience and should be accessible to anyone interested in a window into the true nature of research in mathematics.
About the Speaker: Dr. Bressoud is the DeWitt Wallace Professor of Mathematics at Macalaster College and is President-elect of the MAA. He is also Chair of MAA Committee on Undergraduate Program in Mathematics and also Chair of the MAA Special Interest Group on Teaching Advanced High School Mathematics. Dr. Bressoud has authored the following books: Factorization and Primality Testing, Second Year Calculus from Celestial Mechanics to Special Relativity, A Radical Approach to Real Analysis, Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture, A Course in Computational Number Theory, and A Radical Approach to Lebegue's Theory of Integration.
Wednesday, January 23
Speaker: Rafal Goebel
Dr. Rafal will be giving a talk at 4:00 p.m.
Title: Hybrid dynamical systems -- modeling, stability, and set-valued analysis
Abstract: Hybrid dynamical systems are systems that exhibit behaviors typical of classical continuous-time dynamical systems as well as discrete-time systems. They arise as natural modeling abstractions when describing, for example, mechanical systems with impacts (say, a bouncing ball); control devices operating in various modes (say, a thermostat which is either on or off); robots governed by digital decision-making devices; biological systems; etc.
The talk will give simple examples of hybrid phenomena, motivate the use of hybrid algorithms in control engineering, and present a particular framework for modeling and analysis of hybrid dynamical systems. The
framework will feature hybrid time domains and hybrid inclusions -- a combination of differential inclusions, difference inclusions, and of state constraints on the resulting flows and jumps. Considering inclusions, rather than equations, and abandoning the classical time domains will be motivated by the pursuit of robustness in hybrid control. Representative results in the asymptotic stability theory for hybrid inclusions will be stated. The role of set-valued analysis in the study of hybrid systems will be underlined.
Friday, October 26
Refreshments at 2:30 p.m. in Damen Hall, Room 350
Lecture at 3:00 p.m.
Speaker: Richard Askey, Univ. of Wisconsin-Madison
Title: Ptolemy's theorem, what is it and why do I find it interesting?
Abstract: Ptolemy is probably best known for his attempt to understand astronomy as the Greeks knew it. While his work on this has been replaced, there is a very nice theorem in his great book which is still of interest. One can argue that serious trigonometry started with this theorem. Newton used it, Euler gave a new proof, which combined with Ptolemy's proof can be used to extend his theorem from cyclic quadrilaterals, those whose vertices lie on a circle, to general quadrilaterals. It is a very good problem
to use to teach various ways of learning tools which should be learned in high school geometry, but unfortunately are often not taught now. This and a bit more will be the focus of my talk.
About the Speaker: Professor Askey has more than 180 publications and is one of the world's experts in orthogonal polynomials and special functions. In addition he is one of the leading voices in mathematical education. He is an Honorary Fellow of Indian Academy of Sciences and a Fellow of American Academy of Arts and Science. He received what might be considered his greatest honour in 1999 when he was elected to the National Academy of Sciences.
