Seminars and Colloquia by Series

Scaling limits of Polynomials are fairly universal

Series
Research Horizons Seminar
Time
Wednesday, October 21, 2015 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Prof. Doron Lubinsky – School of Mathematics, Georgia Institute of Technology –

Please Note: Food and Drinks will be provided before the seminar.

In elementary calculus, we learn that (1+z/n)^n has limit exp(z) as n approaches infinity. This type of scaling limit arises in many contexts - from approximation theory to universality limits in random matrices. We discuss some examples.

Regularity theory for surfaces in geometric optics and other Generated Jacobian Equations

Series
PDE Seminar
Time
Tuesday, October 20, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Nestor Guillen – University of Massachusetts at Amherst
The study of reflector surfaces in geometric optics necessitates the analysis of nonlinear equations of Monge-Ampere type. For many important examples (including the near field reflector problem), the equation no longer falls within the scope of optimal transport, but within the class of "Generated Jacobian equations" (GJEs). This class of equations was recently introduced by Trudinger, motivated by problems in geometric optics, however they appear in many others areas (e.g. variations of the Minkowski problem in convex geometry). Under natural assumptions, we prove Holder regularity for the gradient of weak solutions. The results are new in particular for the near-field point source reflector problem, but are applicable for a broad class of GJEs: those satisfying an analogue of the A3-weak condition introduced by Ma, Trudinger and Wang in optimal transport. Joint work with Jun Kitagawa.

Perturbation of linear forms of singular vectors under Gaussian noise

Series
High-Dimensional Phenomena in Statistics and Machine Learning Seminar
Time
Tuesday, October 20, 2015 - 15:00 for 1.5 hours (actually 80 minutes)
Location
Skiles 005
Speaker
Dong Xia – Georgia Inst. of Technology
Let A be a mxn matrix with singular value decomposition A=UDV', where the columns of U are left singular vectors and columns of V are right singular vectors of A. Suppose X is a mxn noise matrix whose entries are i.i.d. Gaussian random variables and consider A'=A+X. Let u_k be the k-th left singular vector of A and u'_k be its counterpart of A'. We develop sharp upper bounds for concentration of linear forms for the right singular vectors of A'.The talk is based on a joint work with Vladimir Koltchinskii.

The infinite topology of the hyperelliptic locus

Series
Geometry Topology Seminar
Time
Monday, October 19, 2015 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Kevin Kordek – Texas A&M
The hyperelliptic Torelli group of a genus g reference surface S_g is the subgroup of the mapping class group whose elements both commute with a fixed hyperelliptic involution of S_g and act trivially on the integral homology of S_g . This group is an important object in geometric topology and group theory, and also in algebraic geometry, where it appears as the fundamental group of the moduli space of genus g hyperelliptic curves with a homology framing. In this talk, we summarize what is known about the (infinite) topology of these moduli spaces, describe a few open problems, and report on some recent partial progress.

Simultaneous Random and Optimized Sources and Detectors for Efficient Optimization in Inverse Problems

Series
Applied and Computational Mathematics Seminar
Time
Monday, October 19, 2015 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Eric de Sturler – Department of Mathematics, Virginia Tech –
In nonlinear inverse problems, we often optimize an objective function involving many sources, where each source requires the solution of a PDE. This leads to the solution of a very large number of large linear systems for each nonlinear function evaluation, and potentially additional systems (for detectors) to evaluate or approximate a Jacobian. We propose a combination of simultaneous random sources and detectors and optimized (for the problem) sources and detectors to drastically reduce the number of systems to be solved. We apply our approach to problems in diffuse optical tomography.This is joint work with Misha Kilmer and Selin Sariaydin.

Self-Avoiding Modes of Motion in a Deterministic Lorentz Lattice Gas

Series
Math Physics Seminar
Time
Friday, October 16, 2015 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Ben Webb – Brigham Young University
We consider the motion of a particle on the two-dimensional hexagonal lattice whose sites are occupied by flipping rotators, which scatter the particle according to a deterministic rule. We find that the particle's trajectory is a self-avoiding walk between returns to its initial position. We show that this behavior is a consequence of the deterministic scattering rule and the particular class of initial scatterer configurations we consider. Since self-avoiding walks are one of the main tools used to model the growth of crystals and polymers, the particle's motion in this class of systems is potentially important for the study of these processes.

Best and random approximation of convex bodies by polytopes

Series
School of Mathematics Colloquium
Time
Thursday, October 15, 2015 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Prof. Dr. Elisabeth Werner – Case Western Reserve University –
How well can a convex body be approximated by a polytope? This is a fundamental question in convex geometry, also in view of applications in many other areas of mathematics and related fields. It often involves side conditions like a prescribed number of vertices, or, more generally, k-dimensional faces and a requirement that the body contains the polytope or vice versa. Accuracy of approximation is often measured in the symmetric difference metric, but other metrics can and have been considered. We will present several results about these issues, mostly related to approximation by “random polytopes”.

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