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Series: Joint ACO and ARC Colloquium

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.

Series: Analysis Working Seminar

We continue our study of Seip's Interpolation Theorem in weighted Bergman spaces. This lecture should cover the necessary direction in the characterization of the Theorem.

Monday, November 2, 2009 - 13:00 ,
Location: Skiles 255 ,
Rustum Choksi ,
Simon Fraser University ,
Organizer:

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.

Series: CDSNS Colloquium

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.

Series: Graph Theory Seminar

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.

Friday, October 30, 2009 - 15:00 ,
Location: Skiles 269 ,
Shea Vela-Vick ,
Columbia University ,
Organizer: John Etnyre

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.

Series: SIAM Student Seminar

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.

Series: Joint ACO and ARC Colloquium

The past 10 years have seen a confluence of research in sparse approximation
amongst computer science, mathematics, and electrical engineering.
Sparse approximation
encompasses a large number of mathematical, algorithmic, and signal
processing problems which all attempt to balance the size of a (linear)
representation of data and the fidelity of that representation. I will
discuss several of the basic algorithmic problems and their solutions,
including connections to streaming algorithms and compressive sensing.

Series: Geometry Topology Seminar

We recall the Schur Weyl duality from representation theory and show how this can be applied to express the colored Jones polynomial of torus knots in an elegant way. We'll then discuss some applications and further extensions of this method.

Series: Analysis Seminar

Given points $z_1,\ldots,z_n$ on a finite open Riemann
surface $R$ and complex scalars $w_1,\ldots,w_n$, the Nevanlinna-Pick
problem is to determine conditions for the existence of a holomorphic
map $f:R\to \mathbb{D}$ such that $f(z_i) = w_i$.
In this talk I will provide some background on the problem, and then
discuss the extremal case. We will try to discuss how a method of
McCullough can be used to provide more qualitative information about
the solution. In particular, we will show that extremal cases are
precisely the ones for which the solution is unique.