Georgia Institute of Technology College of Sciences

We consider (complex) Gaussian analytic functions on a horizontal strip, whose distribution is invariant with respect to horizontal shifts (i.e., "stationary"). Let N(T) be the number of zeroes in [0,T] x [a,b]. First, we present an extension of a result by Wiener, concerning the existence and characterization of the limit N(T)/T as T approaches infinity. Secondly, we characterize the growth of the variance of N(T). We will pose to discuss analogues of these results in a few other settings, such as zeroes of real-analytic Gaussian functions and winding of planar Gaussian functions, pointing out interesting similarities and differences. For the last part, we consider the "persistence probability" (i.e., the probability that a function has no zeroes at all in some region). Here we present results in the real setting, as even this case is yet to be understood. Based in part on joint works with Jeremiah Buckley and Ohad Feldheim.

Mathematical modeling of viruses, such as HIV, has been an extensive area of research over the past two decades. For HIV, some important factors that affect within-host dynamics include: the CTL (Cytotoxic T Lymphocyte) immune response, intra-host diversity, and heterogeneities of the infected cell lifecycle. Motivated by these factors, I consider several extensions of a standard virus model. First, I analyze a cell infection-age structured PDE model with multiple virus strains. The main result is that the single-strain equilibrium corresponding to the virus strain with maximal reproduction number is a global attractor, i.e. competitive exclusion occurs. Next, I investigate the effect of CTL immune response acting at different times in the infected-cell lifecycle based on recent studies demonstrating superior viral clearance efficacy of certain CTL clones that recognize infected cells early in their lifecycle. Interestingly, explicit inclusion of early recognition CTLs can induce oscillatory dynamics and promote coexistence of multiple distinct CTL populations. Finally, I discuss several directions of ongoing modeling work attempting to capture complex HIV-immune system interactions suggested by experimental data.

This talk will tell the story on using simulation for painting. I will tell a few of projects that had simulation and painting involved. One is iPad-based ultra-low-cost real time simulation of old photography process to compute effects that modern day users may find interesting. The other is more full-blown fluid simulation for painting using highest-end GPU. Even with massive processing power of GPU, real time high fidelity painting simulation is hard since computation budget is limited. Basically we should deal with large errors. It may sound odd if someone says that very low-accuracy simulation is interesting - but this is very true. In particular, we tried to pull most pressure effect out from about 10 Jacobi iterations that we could afford. I would like to share my experience on improving fixed number of fixed point iterations.

Dynamical systems have proven to be a useful tool for the design of space missions. For instance, the use of invariant manifolds is now common to design transfer strategies. Solar Sailing is a proposed form of spacecraft propulsion, where large membrane mirrors take advantage of the solar radiation pressure to push the spacecraft. Although the acceleration produced by the radiation pressure is smaller than the one achieved by a traditional spacecraft it is continuous and unlimited. This makes some long term missions more accessible, and opens a wide new range of possible applications that cannot be achieved by a traditional spacecraft. In this presentation we will focus on the dynamics of a Solar sail in a couple of situations. We will introduce this problem focusing on a Solar sail in the Earth-Sun system. In this case, the model used will be the Restricted Three Body Problem (RTBP) plus Solar radiation pressure. The effect of the solar radiation pressure on the RTBP produces a 2D family of "artificial'' equilibria, that can be parametrised by the orientation of the sail. We will describe the dynamics around some of these "artificial'' equilibrium points. We note that, due to the solar radiation pressure, the system is Hamiltonian only for two cases: when the sail is perpendicular to the Sun - Sail line; and when the sail is aligned with the Sun - sail line (i.e., no sail effect). The main tool used to understand the dynamics is the computation of centre manifolds. The second example is the dynamics of a Solar sail close to an asteroid. Note that, in this case, the effect of the sail becomes very relevant due to the low mass of the asteroid. We will use, as a model, a Hill problem plus the effect of the Solar radiation pressure, and we will describe some aspects of the natural dynamics of the sail.