Georgia Institute of Technology College of Sciences

Philip will be presenting topics (and leading discussion on those topics) from Chapter 2 Section 2 of Bounded Analytic Functions.

In the beginning, the basics about random matrix models and some facts about normal random matrices in relation with conformal map- pings will be explained. In the main part we will show that for Gaussian random normal matrices the eigenvalues will fill an elliptically shaped do- main with constant density when the dimension n of the matrices tends to infinity. We will sketch a proof of universality, which is based on orthogonal polynomials and an identity which plays a similar role as the Christoffel- Darboux formula in Hermitian random matrices. Especially we are interested in the density at the boundary where we scale the coordinates with n^(-1/2). We will also consider the off-diagonal part of the kernel and calculate the correlation function. The result will be illustrated by some graphics.

Understanding the mechanisms responsible for the establishment of chronic viral infections is critical to the development of efficient therapeutics and vaccines against highly mutable RNA viruses, such as Hepatitis C (HCV). The mechanism of intra-host viral evolution assumed by most models is based on immune escape via random mutations. However, continuous immune escape does not explain the recent observations of a consistent increase in negative selection during chronic infection and long-term persistence of individual viral variants, which suggests extensive intra-host viral adaptation. This talk explores the role of immune cross-reactivity of viral variants in the establishment of chronic infection and viral intra-host adaptation. Using a computational prediction model for cross-immunoreactivity of viral variants, we show that the level of HCV intra-host adaptation correlates with the rate of cross-immunoreactivity among HCV quasispecies. We analyzed cross-reactivity networks (CRNs) for HCV intra-host variants and found that the structure of CRNs correlates with the type and strength of selection in viral populations. Based on those observations, we developed a mathematical model describing the immunological interaction among RNA viral variants that involves, in addition to neutralization, a non-neutralizing cross-immunoreactivity. The model describes how viral variants escape immune responses and persist, owing to their capability to stimulate non-neutralizing immune responses developed earlier against preceding variants. The model predicts the mechanism of antigenic cooperation among viral variants, which is based on the structure of CRNs. In addition, the model allows to explain previously observed and unexplained phenomenon of reappearance of viral variants: for some chronically infected patients the variants sampled during the acute stage are phylogenetically distant from variants sampled at the earlier years of infection and intermixed with variants sampled 10-20 years later. (Joint work with Y. Khudyakov, Z.Dimitrova, D.Campo and L.Bunimovich)

This talk gives a blowup criteria to the incompressible Navier-Stokes equations in BMO^{-s} on the whole space R^3, which implies the well-known BKM criteria and Serrin criteria. Using the result, we can get the norm of |u(t)|_{\dot{H}^{\frac{1}{2}}} is decreasing function. Our result can obtained by the compensated compactness and Hardy space result of [6] as well as [7].