Graduate Colloquium Winter 2010

January 28, 2010

From Random Matrices to Toeplitz Matrices and back

Prof. Torsten Ehrhardt 

One of the major questions in Random Matrix Theory is the description of the probability distribution of the eigenvalues of a random matrix ensemble.

The study of the linear statistics for the eigenvalues of a random matrix leads to asymptotic problems for Toeplitz matrices and other structured matrices. I will show how operator-theoretic methods can be used to answer these questions, and what this entails for the random matrices.

Both old and new results will be discussed.

February 4, 2010

The Efficient Frontier in Mean-Variance Analysis

Brian Palmer 

One of the primary jobs of an investor is to optimize investment returns, which means to maximize return for a given level of risk. Mean-variance analysis formalizes this task, and yields mathematical tools to deal with this type of optimization. Specifically, investors can fix a level of risk, and find the maximum possible return given this risk level. Finding maximum returns for all possible risk levels produces a set of points described by a differentiable, convex function. This set of points is called the efficient frontier. We will use Lagrange multipliers and a simple but important theorem to find the entire efficient frontier, and talk about uses and consequences that make the frontier worth studying.

February 11, 2010

Norm inflation for incompressible magneto-hydrodynamic system in $\dot{B}_{\infty}^{-1,\infty}$

Mimi Dai 

Recently, Bourgain and Pavlovi\'{c} have proved the ill-posedness of Navier-Stokes equations(NSE) in Besov space $\dot{B}_{\infty}^{-1, \infty}$ in the sense of norm inflation. In their paper, they constructed initial data to NSE using plane waves. Based on the construction of Bourgain and Pavlovi\'{c}, we demonstrate that the solutions to the Cauchy problem for the three dimensional incompressible magneto-hydrodynamics (MHD) system can develop diferent types of norm inflations in \dot{B}_{\infty}^{-1, \infty}$. Particularly the magnetic field can develop norm inflation in short time even when the velocity remains small and vice verse. Efforts are made to be more expository to the ingenious construction of Bourgain and Pavlovi\'{c}.

February 18, 2010

Asymptotically hyperbolic manifolds and AdS/CFT correspondences

Jie Qing, Vice Chair Graduate Programs at UCSC 

In this talk I would like to introduce asymptotically hyperbolic manifolds and some correspondences between the Riemannian geometry of asymptotically hyperbolic manifolds and the conformal geometry of the conformal infinities.