Seminars and Colloquia by Series

Global Smooth Solutions in R^3 to Short Wave-Long Wave Interactions in Magnetohydrodynamics

Series
PDE Seminar
Time
Thursday, October 22, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Hermano Frid – Institute de Matematica Pura e Aplicada (IMPA)
We consider a Benney-type system modeling short wave-long wave interactions in compressible viscous fluids under the influence of a magnetic field. Accordingly, this large system now consists of the compressible MHD equations coupled with a nonlinear Schodinger equation along particle paths. We study the global existence of smooth solutions to the Cauchy problem in R^3 when the initial data are small smooth perturbations of an equilibrium state. An important point here is that, instead of the simpler case having zero as the equilibrium state for the magnetic field, we consider an arbitrary non-zero equilibrium state B for the magnetic field. This is motivated by applications, e.g., Earth's magnetic field, and the lack of invariance of the MHD system with respect to either translations or rotations of the magnetic field. The usual time decay investigation through spectral analysis in this non-zero equilibrium case meets serious difficulties, for the eigenvalues in the frequency space are no longer spherically symmetric. Instead, we employ a recently developed technique of energy estimates involving evolution in negative Besov spaces, and combine it with the particular interplay here between Eulerian and Lagrangian coordinates. This is a joint work with Junxiong Jia and Ronghua Pan.

Random Matrices, the GUE and the distribution of eigenvalues

Series
Other Talks
Time
Wednesday, October 21, 2015 - 17:00 for 1.5 hours (actually 80 minutes)
Location
Skies 006
Speaker
Inoel Popescu – Georgia Tech
This is the fourth meeting in a series of a reading seminars. In this lecture we will analyze the distribution of the eigenvalues of GUE ensembles. We will use Hermite polynomials to get very concrete computations. This way we will recover the semicircular law and we will also discuss a little bit the top eigenvalue.

Small-Time Asymptotic Methods for Levy-Based Jump-Diffusion Models

Series
Mathematical Finance/Financial Engineering Seminar
Time
Wednesday, October 21, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Ruoting Gong – Illinois Institute of Technology
In recent years, small-time asymptotic methods have attracted much attention in mathematical finance. Such asymptotics are especially crucial for jump-diffusion models due to the lack of closed- form formulas and efficient valuation procedures. These methods have been widely developed and applied to diverse areas such as short-time approximations of option prices and implied volatilities, and non-parametric estimations based on high-frequency data. In this talk, I will discuss some results on the small-time asymptotic behavior of some Levy functionals with applications in finance.

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.

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