Seminars and Colloquia by Series

Universal Gaussian fluctuations of non-Hermitian matrix ensembles

Series
Stochastics Seminar
Time
Tuesday, November 3, 2009 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles 255 (Note unusual time and location)
Speaker
Ivan NOURDIN – Paris VI
My aim is to explain how to prove multi-dimensional central limit theorems for the spectral moments (of arbitrary degrees) associated with random matrices with real-valued i.i.d. entries, satisfying some appropriate moment conditions. The techniques I will use rely on a universality principle for the Gaussian Wiener chaos as well as some combinatorial estimates. Unlike other related results in the probabilistic literature, I will not require that the law of the entries has a density with respect to the Lebesgue measure. The talk is based on a joint work with Giovanni Peccati, and use an invariance principle obtained in a joint work with G. P. and Gesine Reinert

The Linearized System for Isometric Embeddings and Its Characteristic Variety

Series
PDE Seminar
Time
Tuesday, November 3, 2009 - 15:05 for 1.5 hours (actually 80 minutes)
Location
Skiles 255
Speaker
Qing Han – University of Notre Dame
We prove a conjecture of Bryant, Griffiths, and Yang concerning the characteristic variety for the determined isometric embedding system. In particular, we show that the characteristic variety is not smooth for any dimension greater than 3. This is accomplished by introducing a smaller yet equivalent linearized system, in an appropriate way, which facilitates analysis of the characteristic variety.

Pricing Catastrophe Put Options Using Methods in Ruin Theory

Series
Mathematical Finance/Financial Engineering Seminar
Time
Tuesday, November 3, 2009 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Sheldon Lin – Department of Statistics, University of Toronto
The discounted penalty function proposed in the seminal paper Gerber and Shiu (1998) has been widely used to analyze the time of ruin, the surplus immediately before ruin and the deficit at ruin of insurance risk models in ruin theory. However, few of its applications can be found beyond, except that Gerber and Landry (1998) explored its use for the pricing of perpetual American put options. In this talk, I will discuss the use of the discounted penalty function and mathematical tools developed for the function for perpetual American catastrophe put options. Assuming that catastrophe losses follow a mixture of Erlang distributions, I will show that an analytical (semi-closed) expression for the price of perpetual American catastrophe put options can be obtained. I will then discuss the fitting of a mixture of Erlang distributions to catastrophe loss data using an EM algorithm.

Counting contingency tables: algorithms and asymptotics

Series
Joint ACO and ARC Colloquium
Time
Monday, November 2, 2009 - 14:00 for 1 hour (actually 50 minutes)
Location
Klaus 1116W
Speaker
Alexander Barvinok – University of Michigan

Please Note: Tea and light refreshments 1:30 in Room 2222. Organizer: Santosh Vempala

I will discuss recent progress on the construction of randomized algorithms for counting non-negative integer matrices with prescribed row and column sums and on finding asymptotic formulas for the number of such matrices (also known as contingency tables). I will also discuss what a random (with respect to the uniform measure) non-negative integer matrix with prescribed row and column sums looks like.

Mathematical Paradigms for Periodic Phase Separation and Self-Assembly of Diblock Copolymers

Series
Applied and Computational Mathematics Seminar
Time
Monday, November 2, 2009 - 13:00 for 30 minutes
Location
Skiles 255
Speaker
Rustum Choksi – Simon Fraser University

Please Note: A density functional theory of Ohta and Kawasaki gives rise to nonlocal perturbations of the well-studied Cahn-Hilliard and isoperimetric variational problems. In this talk, I will discuss these simple but rich variational problems in the context of diblock copolymers. Via a combination of rigorous analysis and numerical simulations, I will attempt to characterize minimizers without any preassigned bias for their geometry.

Energy-driven pattern formation induced by competing short and long-range interactions is ubiquitous in science, and provides a source of many challenging problems in nonlinear analysis. One example is self-assembly of diblock copolymers. Phase separation of the distinct but bonded chains in dibock copolymers gives rise to an amazingly rich class of nanostructures which allow for the synthesis of materials with tailor made mechanical, chemical and electrical properties. Thus one of the main challenges is to describe and predict the self-assembled nanostructure given a set of material parameters.

Stable sets and unstable sets in positive entropy systems

Series
CDSNS Colloquium
Time
Monday, November 2, 2009 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Wen Huang – USTC, China and SoM, Georgia Tech
Stable sets and unstable sets of a dynamical system with positive entropy are investigated. It is shown that in any invertible system with positive entropy, there is a measure-theoretically ?rather big? set such that for any point from the set, the intersection of the closure of the stable set and the closure of the unstable set of the point has positive entropy. Moreover, for several kinds of specific systems, the lower bound of Hausdorff dimension of these sets is estimated. Particularly the lower bound of the Hausdorff dimension of such sets appearing in a positive entropy diffeomorphism on a smooth Riemannian manifold is given in terms of the metric entropy and of Lyapunov exponent.

Color-Critical Graphs Have Logarithmic Circumference

Series
Graph Theory Seminar
Time
Friday, October 30, 2009 - 15:05 for 1.5 hours (actually 80 minutes)
Location
Skiles 255
Speaker
Asaf Shapira – Math and CS, GT
A graph G is k-critical if every proper subgraph of G is (k-1)-colorable, but the graph G itself is not. We prove that every k-critical graph on n vertices has a cycle of length at least logn/100logk, improving a bound of Alon, Krivelevich and Seymour from 2000. Examples of Gallai from 1963 show that this bound is tight (up to a constant depending on k). We thus settle the problem of bounding the minimal circumference of k-critical graphs, raised by Dirac in 1952 and Kelly and Kelly in 1954. This is joint work with Robin Thomas.

Bordered Heegaard-Floer Theory

Series
Geometry Topology Working Seminar
Time
Friday, October 30, 2009 - 15:00 for 2 hours
Location
Skiles 269
Speaker
Shea Vela-Vick – Columbia University
In this talk I will discuss a generalizations and/oo applications of bordered Floer homology. After reviewing the basic definitions and constructions, I will focus either on an application to sutured Floer homology developed by Rumen Zarev, or on applications of the theory to the knot Floer homology. (While it would be good to have attended the other two talks this week, this talk shoudl be independent of them.) This is a 2 hour talk.

Asymptotic behavior of Müntz orthogonal polynomials

Series
SIAM Student Seminar
Time
Friday, October 30, 2009 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Ulfar Stefansson – School of Mathematics, Georgia Tech
After a brief introduction of the theory of orthogonal polynomials, where we touch on some history and applications, we present results on Müntz orthogonal polynomials. Müntz polynomials arise from consideration of the Müntz Theorem, which is a beautiful generalization of the Weierstrass Theorem. We prove a new surprisingly simple representation for the Müntz orthogonal polynomials which holds on the interval of orthogonality, and in particular we get new formulas for some of the classical orthogonal polynomials (e.g. Legendre, Jacobi, Laguerre). This allows us to determine the strong asymptotics on the interval, and the zero spacing behavior follows. We also look at the asymptotic behavior outside the interval, where we apply the method of stationary phase.

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