Mathematics Department Colloquium (2004-2005)

Speaker: Maribel Bueno (Department of Mathematics, William and Mary) 

Title: Polynomial perturbations of bilinear functionals and Hessenberg matrices 
Abstract: This work deals with symmetric and non-symmetric polynomial perturbations of symmetric quasi-definite bilinear functionals. We establish a relation between the Hessenberg matrices associated with the initial and the perturbed functionals using LU and QR factorizations. Moreover we give an explicit algebraic relation between the sequences of orthogonal polynomials associated with both functionals. We also present an extension of the linear symmetrization process to the bilinear case and define the symmetric bilinear functional xL. Finally, we apply our results to functionals whose Hessenberg matrix has a banded power. 

No colloquium talk on October 1. There will be, however, a mathematical talk at our Mathematical-Computational Biology (MCB) seminar on Wednesday, September 29. See the MCB web page for additional information.

October 15, Friday,  3pm in  Jones 131 

Speaker: Laslo Szekely (Department of Mathematics, University of South Carolina) 

Title: The crossing number method 

Abstract

November 5, Friday,  3pm in  Jones 131 

Speaker: Chjan Lim, Department of Mathematical Science, Rensselaer Polytechnic Institute  

Title: Recent advances on Energy-Enstrophy theories of flows on a rotating sphere, and resolution of the low temperature catastrophe in Kraichnan's theory  

Abstract: Literally hundreds of papers have been written in the last 30 years on the so-called Kraichnan's equilibrium statistical mechanics theory for 2d fluid flows, because of its relevance to turbulence, the inverse energy cascade, and the Principle of Selective Decay (Minimum enstrophy) for slightly viscous fluids. The consequences of his theory have been derived in a wide range of flows ranging from simple flows on the plane to multi-layer quasi-geostrophic flows on the rotating sphere. In this talk we prove in mathematical terms that Kraichnan's theory is equivalent to the well-known Gaussian model of theoretical physicists, and thus suffers, as all Gaussian models do, from a low temperature catastrophe where the partition function is not defined. This difficulty is compounded in the case of 2d and 2.5 d fluids because the Green's functions are essentially logarithmic and thus long range. This causes the range of applicability of the Gaussian model to vanish, that is, it works only for inverse temperature beta = 0, in the essential nonextensive continuum or thermodynamic limit. I will propose a resolution of this serious foundational problem, which consists of changing the canonical enstrophy (L_2 norm of vortiicty) constraint in Kraichnan's partition function to a microcanonical constraint. Although a microcanonical constraint is in principle more desirable than a canonical constraint (from physical first principles), it is not often done in practice because the resulting statistical model usually cannot be solved. In the problem of fluids, it turns out that the microcanonical constraint is equivalent to Kac's famous Spherical Model. The second rigorous result in this talk is the exact (closed form) solution of the new microcanonical energy-enstrophy theory. It follows from this exact expression for the partition function and free energy that the new theory is well-defined for all positive and negative temperatures in the same nonextensive continuum limit.  

December 3, Friday,  3pm in  Jones 131 

Speaker: Alexander Tovbis, Department of Mathematics, University of Central Florida  

Title: Method of Riemann-Hilbert Problem for semi-classical limit of the focusing Nonlinear Schroedinger Equation  

Absract: We present the use of Riemann-Hilbert Problem approach to calculationof the leading order term of the solution of the focusing Nonlinear (cubic) Schroedinger Equation (NLS) in the semi-classical limit for a certain one-parameter family of initial conditions. This family contains both solitons and pure radiation. We also calculate the long-term asymptotics of this limit.

January 7, Friday,  3pm in  Jones 131 

Speaker: Brian Sutton, Department of Mathematics, MIT  

Title: The stochastic operator approach to random matrices  

February 9, Wednesday,  3pm in  Jones 131 

Speaker: Michael Doob, University of Manitoba  

Title: Ramanujan graphs from a combinatorial viewpoint  

Absract: A Ramanujan graph is defined by a bound on the second largest eigenvalue of its spectrum. These graphs initially arose in the study of expanders in communication networks, but more recently have become of interest to number theorists because of some clever (and difficult) constructions. We look at some combinatorial implications of these constructions, which indicate further directions of research.

February 21, Monday,  3pm in  Jones 131 

Speaker: Roger Barnard, Texas Tech  

Title: How far can you deform a disk under a convex map?

Absract: In this talk we will discuss how we apply variational techniques and special function theory to verify some conjectures of D. Minda's and of C. Pommerenke's on the sharp bound for the Schwarzian derivative of hyperbolically convex maps. This completes the classification of the extremal domains for the Schwarzian in all three classical geometries hence answering the question first posed in the 50's as to how far one can distort a disk under a convex map in Euclidean, spherical and hyperbolic geometries.

March 18, Friday,  3pm in  Jones 131 

Speaker: Ren-Cang Li, University of Kentucky  

Title: Conditioning of Vandermonde Matrices, and Krylov Subspace Methods

Absract: Real Vandermonde matrices are notoriously known for its ill-conditioning, but their asymptotically optimal lower bounds on their condition numbers had not been found until recently, independently by Beckermann and myself. Rectangular Vandermonde matrices play a crucial role in the convergence analysis of Krylov subspace methods for linear systems and eigenvalue problems -- a fact that has not been exploited so far. In a way precisely it is the ill-conditioning of (real) Vandermonde Matrices that contributes to the fast convergence of CG, MINRES, and symmetric Lanczos method. In this talk, we will present various new results on asymptotically optimal lower bounds on the condition numbers of real Vandermonde matrices and sharpness of the existing error bounds of CG, MINRES, and Lanczos method.