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.
Wednesday, April 11, 2012 - 12:05 , Location: Skiles 005 , John McCuan , Georgia Tech , Organizer:
Classical mathematical capillarity theory has as its foundation variational methods introduced by Gauss. There was a heuristic explanation given earlier by Thomas Young, and his explanations did have quantitative scientific content. Due partially to their simplistic nature, the explanations of Young live on today in engineering textbooks, though in certain cases it has been pointed out that they lead to anomolous predictions (which are effectively avoided in the Gaussian variational framework). I will discuss a fundamentally new direction in mathematical capillarity which is motivated by an effort to harmonize the heuristic and rigorous elements of the theory and has other important applications as well.
Two applications of Riemannian geometry: Thermal stresses in solid mechanics and density preserving maps in machine learning.Wednesday, April 4, 2012 - 12:05 , Location: Skiles 005 , Arkadaş Özakın , Georgia Tech Research Institute , Organizer:
I will present two "real life" applications of Riemannian geometry,one in the field of continuum mechanics, another in the field ofmachine learning (nonlinear dimensionality reduction). I will providequick introductions in order to make the talk accessible to anaudience with no background in either field.
Wednesday, March 28, 2012 - 12:05 , Location: Skiles 005 , Anton Leykin , Georgia Tech , Organizer:
This talk will traverse several topics in singularity theory, algebraic analysis, complex analysis, algebraic geometry, and statistics. I will outline effective methods to compute the log canonical threshold, a birational invariant of an algebraic variety, as well as its potential statistical applications.
Wednesday, March 7, 2012 - 12:05 , Location: Skiles 005 , Robin Thomas , Georgia Tech , Organizer:
I will present a construction of a non-measurable set using the fundamental fact that a graph with no odd cycles is 2-colorable. That will not take very long, even though I will prove everything from first principles. In the rest of the time I will discuss the Axiom of Choice and some unprovable statements. The talk should be accessible to undergraduates.
Wednesday, February 22, 2012 - 12:05 , Location: Skiles 005 , Prasad Tetali , Georgia Tech , Organizer:
Following exciting developments in the continuous setting of manifolds (and other geodesic spaces), in joint works with various collaborators, I have explored discrete analogs of the interconnection between several functional and isoperimetric inequalities in discrete spaces. Such inequalities include concentration, transportation, modified versions of the logarithmic Sobolev inequality, and (most recently) displacement convexity. I will attempt to motivate and review some of these connections and illustrate with examples. Time permitting, computational aspects of the underlying functional constants and other open problems will also be mentioned.