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| Monday 27 |
3:10-4 pm, room 1432. Graph Theory and Combinatorics Seminar. Paul Edelman, Vanderbilt University. The Inverse Banzhaf Problem. Let F be a family of subsets of the ground set [n]={1,2,...,n}. For
each i in [n] we let p(F,i) be the number of pairs of subsets that differ
in the element i and exactly one of them is in F. We interpret p(F,i) as
the influence of that element. The normalized Banzhaf vector of F,
denoted B(F), is the vector (B(F,1),...,B(F,n)), where B(F,i)= p(F,i)/p(F)
and p(F) is the sum of the p(F,i). The Banzhaf vector has been studied in
the context of measuring voting power in voting games as well as in
Boolean circuit theory. In this paper we ask investigate which
non-negative vectors of sum 1 can be closely approximated by Banzhaf
vectors of simplicial complexes. In particular, we show that if a vector
has most of its weight concentrated in k < n coordinates, then it must be
essentially the Banzhaf vector of some k-complex. This is joint work with
Noga Alon.
4:10-5 pm, room 1431. NCGOA Research Training Group Seminar. Bogdan Nica, Vanderbilt University. Elementary generation of the special linear group II. |
| Tuesday 28 |
3:20 pm, room 1425. Graduate Student Tea.
4:10-5 pm, room 1432. Noncommutative Geometry Seminar. Dan Ramras, Vanderbilt University. An introduction to Quillen's algebraic K-theory. In the early 70's, Quillen discovered a topological construction which unified the lower algebraic K-groups defined by Grothendieck (K_0), Whitehead (K_1), and Milnor (K_2) and produced a notion of higher algebraic K-theory. I'll describe Quillen's approach to higher K-theory for rings. In particular, I'll describe his Q-construction, and I'll explain why it gives the correct definition of K_0, the group of formal differences between projective modules. I'll also describe some of the theorems Quillen proved using this approach. This will be an introductory talk, and I will try to keep the formal nonsense to a minimum. 4:10-5:00 pm, room 1312. Computational Analysis Seminar. Rick Chartrand, Los Alamos National Laboratory. Nonconvex compressive sensing: getting the most from very little information (and the other way around). In this talk we'll look at the exciting, recent results showing that most images and other signals can be reconstructed from much less information than previously thought possible, using simple, efficient algorithms. A consequence has been the explosive growth of the new field known as compressive sensing, so called because the results show how a small number of measurements of a signal can be regarded as tantamount to a compression of that signal. The many potential applications include reducing exposure time in medical imaging, sensing devices that can collect much less data in the first place instead of collecting and then compressing, getting reconstructions from what seems like insufficient data (such as EEG), and very simple compression methods that are effective for streaming data and preserve nonlinear geometry. We'll see how replacing the convex optimization problem typically used in this field with a nonconvex variant has the effect of reducing still further the number of measurements needed to reconstruct a signal. A very surprising result is that a simple algorithm, designed only for finding one of the many local minima of the optimization problem, typically finds the global minimum. Understanding this is an interesting and challenging theoretical problem. We'll see examples, and discuss algorithms, theory, and applications. 4:30-5:30 pm, room 1308. Universal Algebra and Logic Seminar. Constantine Tsinakis, Vanderbilt University. Universal Algebra for the Working Mathematician II. The title of the talk captures its aims. My intention is to provide an account of those fundamental results in the area that every well-educated research mathematician would find useful in his/ her research. One of the aims of universal algebra is to study features common to many familiar algebraic systems, such as groups, rings, lattices, etc. Such a study places a number of algebraic notions in their proper setting, reveals connections of seemingly unrelated concepts, and uses the higher level of abstraction to apply these results to entirely new situations. A central theme in this area is that of a variety or an equational class. 7-8 pm, room 1206. Undergraduate Seminar in Mathematics. Matt Calef, Vanderbilt University. Across the Eighth Dimension (Buckaroo Bonzai!) While the term dimension is used regularly it has many different definitions not all of which agree. An intuitive understanding is that the dimension is the maximum number of mutually perpendicular directions. However, dimension can be meaningful in settings where the notion of direction is not. The talk will start by considering the dimension of objects familiar to first year calculus students and then move on to examining dimension in more involved settings. We shall see that the questions: Can the dimension be infinite? and Can the dimension be non-integer? are both answered in the affirmative. Free pizza. |
| Wednesday 29 | 4:10 pm, room 1310. Topology & Group Theory Seminar. Yves de Cornulier, Institute of Mathematical Research of Rennes, Rennes. On the Dehn function of some solvable groups. We compute the Dehn function for a class of metabelian groups. In particular, the solvable Baumslag-Solitar groups can be embedded into finitely generated metabelian groups with quadratic Dehn function. (Joint work with R. Tessera) |
| Thursday 30 | 4:10-5 pm, room 5211. Colloquium. Roman Vershynin, University of Michigan. A geometric view of random matrices. We examine a classical object in random matrix theory - rectangular matrices with random independent entries - from a geometric point of view. How do these matrices act as linear operators in high dimensional spaces? How do they transform convex sets? It often turns out that random matrices are the best transformations one can hope for, with no deterministic constructions known of similar quality. We shall survey a rich history of old and recent developments, focusing on the recent progress on the invertibility problem for random matrices. Tea at 3:30 pm in SC 1425. |
| Friday 31 | 4:10 pm, room 1307. Partial Differential Equations Seminar. Mike Frazier, University of Tennessee, Knoxville. Estimates for Green's functions of Schrodinger operators. We consider the inhomogeneous, time-independent Schrodinger equation. Under certain conditions on the potential, we obtain global lower and upper exponential estimates for the Green's function of the Schrodinger operator in terms of the first and second iterates of the Green's function for the Laplacian. The estimates hold on the whole space and for a very general class of domains. The results for Schrodinger operators are a consequence of a more general result. If T is a bounded linear operator on L^2 (\mu) with norm less than one, then I-T has an inverse given by a Neumann series. Suppose T is represented by integration against a symmetric kernel K(x,y). Under the condition that the reciprocal of K is a quasimetric, we obtain global exponential bounds (both lower and upper, but with different constants) for the kernel of the inverse of I-T. Our methods also apply to operators with fractional potential replacing the Laplacian. These operators relate to alpha-stable Levy processes in the same way that the Laplacian relates to Brownian motion. (Joint work with Fedor Nazarov and Igor Verbitsky) |