Departmental Colloquium

Archival event entry migrated from the legacy colloquium website.

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.

Archival event entry migrated from the legacy colloquium website.

Archival event entry migrated from the legacy colloquium website.

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.

We will discuss non-positively curved cubical complexes and their relationship to various topics in geometric group theory.

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.

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.

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.

The goal of this talk is to present a new noncommutative tool in the study of the subfactor theory. The theory of von Neumann algebras was introduced by Murray and von Neumann as the mathematical foundation of quantum mechanics. Every von Neumann algebra can be decomposed as a direct "sum" of factors. Before the `80's the theory is focused on the classification of factors. Subfactor theory became the mainstream after Vaughan Jones's work. Subfactor theory studies the position of a subfactor sitting inside the ambient factor. The standard invariant of the inclusion is a complete invariant in "good" cases and can be packed pictorially as the associated planar algebra. The simplest example is the Temperley-Lieb algebra, which gives a representation of braid group. The invariant in this case yields a knot invariant, the celebrated Jones polynomial. Irreducible inclusions are the most benign ones, which allow a classification result in term of planar algebra. A II 1 hyperfinite factor can be approximated by finite dimensional C * algebras and has a unique normalized trace. CAR (canonical anticommutation relations) algebra is a typical example. In this talk, I will present a new technique of constructing irreducible hyperfinite II 1 inclusions. The example is a series of inclusions of II 1 factors inside the Temperley-Lieb algebra R, R + P 1 + P 2 +... + P n +... with the property: The "size" of the subfactor P n compared to R decays exponentially, but each inclusion P n R remains irreducible. One application is an extension of Temperley-Lieb algebra with extremality. When the Jones index is around 4, extremality is equivalent to irreducibility. It is a long standing problem (so-called gap conjecture ) in subfactor theory whether Jones index is "quantized" when it is bigger than 4. The above result represents a progress towards this direction.