Departmental Colloquium
Fall 2005
Regular Day: Thursday
Regular Time: 4:00PM - 5:00PM
Regular Location: JWB 335
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September 22 |
István Berkes
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An infinite sequence in (0, 1) is called uniformly distributed in the Weyl sense if any subinterval (a,b) of (0,1) contains, in an asymptotic sense, a fraction b-a of the terms of the sequence. Such sequences play an important role in computer science (random number generation, Monte Carlo integration, simulation), as well as in analysis and number theory. How close the distribution of a sequence is to uniform is measured by a quantity called discrepancy. The mathematical "prototype" for a uniformly distributed sequence is {nx} and its suitable subsequences, where x is an irrational number. Computing the discrepancy of such sequences is a long open problem of analysis, the solution of which is known only in a few special cases. The purpose of our talk is to prove that, in a certain sense, "almost all" such sequences have exactly the same discrepancy behavior. The probabilistic method of the proof will also be used to solve some open problems in analysis related to the classical Khinchin conjecture.
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September 29 Distinguished Lecture Series Undergraduate Colloquium |
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Archival event entry migrated from the legacy colloquium website.
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September 30 Distinguished Lecture Series Colloquium |
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The problem of classifying n x n matrices over a field k has two aspects. One is arithemetic: any degree n extension field K of k can be embedded in nxn matrices, in a way canonical up to conjugacy. In terms of linear algebra, the corresponding matrices have their eigenvalues in K. For this reason, the arithemtic of k affects their conjugacy classes. I will ignore this aspect entirely. The second aspect is independent of field; one could call it purely "algebraic," if that word is divorced from arithmetic. There are nonzero n x n matrices all of whose eigenvalues are zero. These are nilpotent matrices. Any nilpotent nxn matrix is conjugate to one in Jordan normal form, and in this way conjugacy classes of nilpotent matrices are in bijection with the partitions of n. More than a hundred years ago, Frobenius discovered exactly the same set (partitions of n ) parameterizes the irreducible representations of the symmetric group S n. Since that time, there has been a tremendous amount of work aimed at using information about S n and its representations (which are a part of combinatorics and finite mathematics) to study GL(n) and its representations (which are part of algebraic geometry, arithmetic and analysis.) I will describe two examples of this work. The first, due to Green in 1955, shows how to use the symmetric group to understand certain geometrically natural representations of GL(n) over a finite field. He shows that these representations in GL(n) are " q -analogues" of certain natural symmetric group representations, and that all of these representations decompose in exactly the same way. The second example concerns the algebraic variety N of all n x n nilpotent matrices over an algebraically closed field. Fifteen years ago, Lusztig conjectured a very precise relationship between the GL(n) invariant coherent sheaves on N, and some combinatorics related to the symmetric group S n (and its semidirect product with Z n.) Lusztig's conjecture has been proven by Bezrukavnikov and Achar.
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October 1 Distinguished Lecture Series Lie Groups Seminar |
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Archival event entry migrated from the legacy colloquium website.
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October 7 Fall Break Holiday |
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Archival event entry migrated from the legacy colloquium website.
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October 20 |
Walter Neumann
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This talk will describe the status of Hilbert's third problem on equidecomposability of polytopes, whose solution by Dehn in 1900 led to refinements that remain unsolved and that have interesting links with invariants of 3-manifolds.
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October 21 Special Colloquium |
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Although global projections such as truncated Fourier series yield exponentially close approximations for smooth functions, a single discontinuity introduces O(1) spurious oscillations, Gibbs' Phenomena, and reduces the high order convergence rate to first order. A family of filters with different properties have been developed over the last century to reduce the effects of the Gibbs phenomenon; however, which filter to use for a given application has remained largely heuristic. In the first half of the talk I construct a spatially adaptive filter which is shown to achieve optimal (exponential) accuracy for this class of methods, and as a result definitively resolves the question of filter selection. Second I consider a problem in modern communication and signal processing, the recovery of a bandlimited signal from its bunched samples. In many emerging applications the uniform sampling required for the classical Shannon sampling theorem is not realizable, and instead bunched sampling is utilized. Here I present an efficient and robust high order algorithm for the "bunched" sampling structure, with the same exponential accuracy as can be achieved with uniform sampling. Portions of these research projects were conducted in collaboration with Eitan Tadmor and Thomas Strohmer.
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October 27 |
Eric Sharpe
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String theory is a proposed theory of physics, which has turned out to have numerous interactions with mathematics. In this talk I will review some examples of mathematics motivated by this bit of physics, which are studied by several department members. I will begin by reviewing ``mirror symmetry,'' whose understanding brought about an industry in the algebraic geometry community, and then describe some of its modern-day spinoffs, including ``(0,2) mirror symmetry'' and ``homological mirror symmetry.''
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October 28 Departmental Colloquium |
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We will discuss non-positively curved cubical complexes and their relationship to various topics in geometric group theory. * Return to top *
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November 3 |
Alexander Balk
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I will talk about the crucial role of conservation laws in turbulence. The talk consists of three parts: 1. What is the problem of turbulence? Why is it important? Why is it a challenge? 2. The conservation laws determine the main featutres of turbulence. 3. In the last part I will present my new results explaining the widely observed phenomenon of the formations of zonal jets in the turbulence of the ocean and atmosphere.
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November 4 Departmental Colloquium |
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The Orbiter Columbia disintegrated on re-entry on Saturday, February 1, 2003. The next Monday, Southwest Research Institute was contacted about performing tests (and later analysis) to determine the plausibility of a foam impact on the left wing being the cause of the loss of the vehicle. This talk discusses the work that was performed at SwRI during the Columbia investigation, sponsored by both NASA and the Columbia Accident Investigation Board. The work included test impacts into both thermal tiles and into the reinforced carbon-carbon leading edge panels. Both impact scenarios were also computationally modeled. The presentation will include high-speed video of foam impact tests and computer animations of numerical simulations of impact. The work showed that a foam impact on the leading edge of the wing could produce significant damage. These results, combined with the forensic work of the investigation, allowed the conclusion that Columbia was lost due to the impact of external-tank insulating foam near the left wing's RCC panel 8 during ascent. More details on SwRI's role in the investigation (and additional photographs) can be found at http://www.swri.edu/3pubs/ttoday/fall03/LeadingEdge.htm.
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November 10 |
Dave Keyes
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The Terascale Optimal PDE Simulations (TOPS) project is sponsored by the Department of Energy to research, implement, deploy, maintain, and support in collaborations with scientific users a collection of open-source, scalable solvers for the discretized problems in such applications as fusion reactor modeling and design, climate, and combustion. Optimal complexity methods, such as multigrid/multilevel preconditioners, keep the time spent in dominant algebraic kernels linear as the applications scale on massively parallel computers. Krylov accelerators and Jacobian-free variants of Newton's method, as appropriate, are wrapped around the multilevel methods to deliver robustness in multirate, multiscale coupled systems, which are solved either implicitly or in more traditional forms of operator splitting. The TOPS software framework is being extended beyond direct computational simulation to computational optimization, including design, control, and inverse problems. We outline the capabilities TOPS offers to high-end simulations generally, and illustrate on applications in magnetically confined fusion energy, as the U.S. gears up for participation in the International Thermonuclear Experimental Reactor (ITER) consortium, the ultimate goal of which is abundant exportable energy production capability, outside of the planetary carbon cycle.
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November 11 Departmental Colloquium |
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I will discuss a new type of cohomology, depending on a real parameter q, of a cell complex associated to an infinite Coxeter group W. As q varies these cohomology groups interpolate between ordinary cohomology and cohomology with compact supports. Although these groups are Hilbert spaces (usually infinite dimensional), they can be assigned a "dimension" which is a nonnegative real number. These "dimensions" vary continuously with the parameter q. When q is an integer, they give the L 2 Betti numbers of any regular building of type W and thickness q.
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November 17 |
Chandreshekar Khare
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In the early 1970's, Serre made a conjecture about mod p Galois representations that has been very influential. I will talk about recent progress towards the resolution of this conjecture.
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November 18 Departmental Colloquium |
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Recent experiments in the Courant Institute Applied Math Lab have shown fascinating and subtle interactions between fluids and bodies moving through them, and have suggested that fluidic response to body motions can change dramatically with the "forcing Reynolds number," and can result in body locomotion. In joint work with S. Alben, I will discuss the rich dynamics possible from an oscillated simple body that interacts with a surrounding viscous fluid. We show that nontrivial dynamics results from a classical symmetry breaking instability of body/flow interaction, yielding new periodic, quasi-periodic, and apparently chaotic dynamical states. We show further that in broad parameter regimes, unidirectional coherent locomotion of the body emerges as an attracting state of the system.
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December 1 |
Gang Tian
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Archival event entry migrated from the legacy colloquium website.
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