Georgia Institute of Technology College of Sciences

The emerging threat of a human pandemic caused by high-pathogenic H5N1 avian in uenza virus magnifies the need for controlling the incidence of H5N1 in domestic bird populations. The two most widely used control measures in poultry are culling and vaccination. In this talk, I will discuss mathematical models of avian in uenza in poultry which incorporate culling and vaccination. First, we consider an ODE model to understand the dynamics of avian influenza under different culling approaches. Under certain conditions, complex dynamical behavior such as bistability is observed and analyzed. Next, we model vaccination of poultry by formulating a coupled ODE-PDE model which takes into account vaccine-induced asymptomatic infection. In this study, the model can exhibit the "silent spread" of the disease through asymptomatic infection. We analytically and numerically demonstrate that vaccination can paradoxically increase the total number of infected when the efficacy is not sufficiently high.

We discuss an approach to dyadic lattices (and their applications to harmonic analysis) presented by Lerner and Nazarov in their manuscript, Intutive Dyadic Calculus.

Given their ubiquity in physics, chemistry and materialsciences, cluster nucleation and growth have been extensively studied,often assuming infinitely large numbers of buildingblocks and unbounded cluster sizes. These assumptions lead to theuse of mass-action, mean field descriptions such as the well knownBecker Doering equations. In cellular biology, however, nucleationevents often take place in confined spaces, with a finite number ofcomponents, so that discrete and stochastic effects must be takeninto account. In this talk we examine finite sized homogeneousnucleation by considering a fully stochastic master equation, solvedvia Monte-Carlo simulations and via analytical insight. We findstriking differences between the mean cluster sizes obtained from ourdiscrete, stochastic treatment and those predicted by mean fieldones. We also study first assembly times and compare results obtained from processes where only monomer attachment anddetachment are allowed to those obtained from general coagulation-fragmentationevents between clusters of any size.

Construction of pseudo-Anosov elements of mapping class groups of surfaces is a non-trivial task. In 1988, Penner gave a very general construction of pseudo-Anosov mapping classes, and he conjectured that all pseudo-Anosov mapping classes arise this way up to finite power. This conjecture was known to be true on some simple surfaces, including the torus. In joint work with Hyunshik Shin, we resolve this conjecture for all surfaces.