Wednesday, November 14, 2012 - 12:05 , Location: Skiles 005 , Matt Baker , Georgia Tech, School of Math , Organizer: Robert Krone
I will discuss how one can solve certain concrete problems in number theory, for example the Diophantine equation 2x^2 + 1 = 3^m, using p-adic analysis. No previous knowledge of p-adic numbers will be assumed. If time permits, I will discuss how similar p-adic analytic methods can be used to prove the famous Skolem-Mahler-Lech theorem: If a_n is a sequence of complex numbers satisfying some finite-order linear recurrence, then for any complex number b there are only finitely many n for which a_n = b.
Wednesday, November 7, 2012 - 12:05 , Location: Skiles 005 , Santosh Vempala , Georgia Tech, College of Computing , Organizer: Robert Krone
The hyperplane conjecture of Kannan, Lovasz and Simonovits asserts that the isoperimetric constant of a logconcave measure (minimum surface to volume ratio over all subsets of measure at most half) is approximated by a halfspace to within an absolute constant factor. I will describe the motivation, implications and some developments around the conjecture and an approach to resolving it (which does not seem entirely ridiculous).
Wednesday, October 31, 2012 - 12:05 , Location: Skiles 005 , John Etnyre , Georgia Tech, School of Math , Organizer: Robert Krone
Contact geometry is a beautiful subject that has important interactions with topology in dimension three. In this talk I will give a brief introduction to contact geometry and discuss its interactions with Riemannian geometry. In particular I will discuss a contact geometry analog of the famous sphere theorem and more generally indicate how the curvature of a Riemannian metric can influence properties of a contact structure adapted to it.
Wednesday, October 24, 2012 - 12:05 , Location: Skiles 005 , Brett Wick , Georgia Tech: School of Math , Organizer: Robert Krone
In this talk we will connect functional analysis and analytic function theory by studying the compact linear operators on Bergman spaces. In particular, we will show how it is possible to obtain a characterization of the compact operators in terms of more geometric information associated to the function spaces. We will also point to several interesting lines of inquiry that are connected to the problems in this talk. This talk will be self-contained and accessible to any mathematics graduate student.
Wednesday, October 17, 2012 - 12:05 , Location: Skiles 005 , Mohammad Ghomi , Georgia Tech - School of Math , Organizer: Robert Krone
One of the most outstanding problems in differential geometry is concerned with flexibility of closed surface in Euclidean 3-space: Is it possible to continuously deform a smooth closed surface without changing its intrinsic metric structure? In this talk I will give a quick survey of known results in this area, which is primarily concerned with convex surfaces, and outline a program for studying the general case.
Wednesday, October 3, 2012 - 12:05 , Location: Skiles 005 , Zhiwu Lin , Georgia Tech, School of Math , Organizer: Robert Krone
Consider electrostatic plasmas described by Vlasov-Poisson with a fixed ion background. In 1946, Landau discovered the linear decay of electric field near a stable homogeneous state. This phenomena has been puzzling since the Vlasov-Poisson system is time reversible and non-dissipative. The nonlinear Landau damping was proved for analytic perturbations by Mouhot and Villani in 2009, but for general perturbations it is still largely open. I will discuss some recent results with C. Zeng on the failure of nonlinear daming in low regularity neighborhoods and a regularity threshold for the existence of nontrivial invariant structures near homogeneous states. A related problem to be discussed is nonlinear inviscid damping of Couette flow, for which the linear decay was first observed by Orr in 1907.
Wednesday, September 26, 2012 - 12:05 , Location: Skiles 005 , Igor Belegradek , Georgia Tech, School of Math , Organizer: Robert Krone
In the talk we will start from examples of open surfaces, such as the complex plane minus a Cantor set, review their classification, and then move to higher dimensions, where we discuss ends of manifolds in the topological setting, and finally in the geometric setting under the assumption of nonpositive curvature.
Wednesday, September 12, 2012 - 12:05 , Location: Skiles 005 , Doron Lubinsky , School of Mathematics, Georgia Tech , Organizer: Robert Krone
Orthogonal polynomials turn out to be a useful tool in analyzing random matrices. We present some old and new aspects.
Wednesday, September 5, 2012 - 13:05 , Location: Skiles 005 , William Trotter , School of Mathematics, Georgia Tech , Organizer: Robert Krone
We survey research spanning more than 20 years on what starts out to be a very simple problem: Representing a poset as the inclusion order of circular disks in the plane. More generally, we can speak of spherical orders, i.e., posets which are inclusion orders of balls in R^d for some d. Surprising enough, there are finite posets which are not sphere orders. Quite recently, some elegant results have been obtained for circle orders, lending more interest to the many open problems that remain.
Wednesday, April 18, 2012 - 12:05 , Location: Skiles 005 , Vladimir Koltchinskii , Georgia Tech , Organizer:
Recently, there has been a lot of interest in estimation of sparse vectors in high-dimensional spaces and large low rank matrices based on a finite number of measurements of randomly picked linear functionals of these vectors/matrices. Such problems are very basic in several areas (high-dimensional statistics, compressed sensing, quantum state tomography, etc). The existing methods are based on fitting the vectors (or the matrices) to the data using least squares with carefully designed complexity penalties based on the $\ell_1$-norm in the case of vectors and on the nuclear norm in the case of matrices. Proving error bounds for such methods that hold with a guaranteed probability is based on several tools from high-dimensional probability that will be also discussed.