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WEEKLY CALENDAR |
| Monday 17 |
3:10 pm, room 1431. Graph Theory and Combinatorics Seminar. Paul Edelman, Vanderbilt University.
The Inverse Banzhaf Problem. Let S be a simplicial complex on the ground set {1,2,...,n} =[n]. For each i in [n] let B(S,i) be the number of subsets A in S, which do not contain i, such that A+i is not in S. That is, B(S,i) is the number of times adding i to a set in S produces a set not in S. B(S)=(B(S,1), B(S,2),...,B(S,n)) is called the Banzhaf index of S, and after normalizing it to get a sum of 1 we have the NB(s), the normalized Banzhaf index. The Normalized Banzhaf index has been studied in the context of voting games as well as in Boolean circuit theory. In this talk I will consider the inverse problem. Given a nonnegative vector v, of sum 1, when can we find a simplicial complex S so that NB(S) is a good approximation to v? There is virtually nothing known on this subject. I will present some preliminary results that show that one can never well-approximate vectors on the boundary of the standard simplex.
4:10-5:30 pm, room 1432. Subfactor Seminar. Alan Wiggins, Vanderbilt University. A new construction of subfactors from random matrix theory VI. |
| Tuesday 18 | 7-8 pm, room 1206. Undergraduate Seminar in Mathematics. Derek Bruff, Vanderbilt University. The Incredible Shrinking Data. In Clarksburg, West Virginia, the FBI maintains a database of over 200 million sets of fingerprints. That's about 4000 hard drives full of information! How does the FBI store all this data? They shrink it, of course, just as you might do to a song when you store it on your computer as an MP3 or to a photo you take with your digital camera. In this talk, we'll look at wavelets, the mathematical tools behind the data compression used by MP3s, digital photos, and the FBI. Free pizza. |
| Wednesday 19 | 4:10 pm, room 1310. Topology & Group Theory Seminar. Dan Ramras, Vanderbilt University. Deformation K-theory and the stable moduli space of flat connections. Deformation K-theory studies the homotopy orbit spaces Hom(G, U(n))_hU(n) of unitary representation spaces. Work of Tyler Lawson provides a surprising relationship between these homotopy orbit spaces and the quotient spaces Hom(G, U(n))/U(n). Lawson used this relationship to show that for finitely generated free groups F_k, Hom(F_k, U)/U is homotopy equivalent to the k-dimensional torus (S1)^k. The latter space may also be viewed as the infinite symmetric product of the classifying space B(F_k). I'll explain how to prove an analogous result for for fundamental groups of surfaces M, in which case Hom(\pi_1 M, U(n))/U(n) is the moduli space of flat unitary connections on bundles over M. |
| Thursday 20 | 4:10-5 pm, room 5211. Colloquium. Bruce Kleiner, Yale University . A new proof of Gromov's theorem on groups of polynomial growth. In 1981 Gromov showed that any finitely generated group of polynomial growth contains a finite index nilpotent subgroup. This was a landmark paper in several respects. The proof was based on the idea that one can take a sequence of rescalings of an infinite group G, pass to a limiting metric space, and apply deep results about the structure of locally compact groups to draw conclusions about the original group G. In the process, the paper introduced Gromov-Hausdorff convergence, initiated the subject of geometric group theory, and gave the first application of the Montgomery-Zippin solution to Hilbert's fifth problem (and subsequent extensions due to Yamabe). The purpose of the lecture is to give a new, much shorter, proof of Gromov's theorem. The main step involves showing that any infinite group of polynomial growth admits a finite dimensional linear representation with infinite image. We establish this using harmonic maps, thereby avoiding the Montgomery-Zippin-Yamabe theory of locally compact groups which was used in Gromov's original proof. I will explain the proof in a manner accessible to a broad audience. Tea at 3:30 pm in SC 1425. |
| Friday 21 |
4:10-5:30 pm, room 1432.
Special Subfactor Seminar. Cyril Houdayer, UCLA. Prime factors and amalgamated free products. I will show that a non-amenable II_1 factor arising as an amalgamated free product over an abelian von Neumann algebra is prime. I will moreover discuss some applications and generalizations of this result. This is joint work with Ionut Chifan.
4:10-5:00 pm, room 1307. Partial Differential Equations Seminar. David Hoff, Indiana University. Analyticity in time and backward uniqueness of weak solutions of the Navier-Stokes equations of multidimensional, compressible flow. We prove that solutions of the Navier-Stokes equations of three-dimensional, compressible flow, restricted to fluid-particle trajectories, can be extended as analytic functions of complex time. One important corollary is backwards uniqueness: if two such solutions agree at a given time, then they must agree at all previous times. Additionally, analyticity yields sharp estimates for the time derivatives of arbitrary order of solutions along particle trajectories. I'm going to integrate into the talk something like a "pretalk," in an attempt to motivate the more technical material and to make things accessible to general analysis grad students. All are welcome. |