Georgia Institute of Technology College of Sciences

Spatially discrete stochastic models have been implemented to analyze cooperative behavior in a variety of biological, ecological, sociological, physical, and chemical systems. In these models, species of different types, or individuals in different states, reside at the sites of a periodic spatial grid. These sites change or switch state according to specific rules (reflecting birth or death, migration, infection, etc.) In this talk, we consider a spatial epidemic model where a population of sick or healthy individual resides on an infinite square lattice. Sick individuals spontaneously recover at rate *p*, and healthy individual become infected at rate O(1) if they have two or more sick neighbors. As *p* increases, the model exhibits a discontinuous transition from an infected to an all healthy state. Relative stability of the two states is assessed by exploring the propagation of planar interfaces separating them (i.e., planar waves of infection or recovery). We find that the condition for equistability or coexistence of the two states (i.e., stationarity of the interface) depends on orientation of the interface. We analyze this stochastic model by applying truncation approximations to the exact master equations describing the evolution of spatially non-uniform states. We thereby obtain a set of discrete (or lattice) reaction-diffusion type equations amenable to numerical analysis.

Many physical models without dissipation can be written in a Hamiltonian form. For example, nonlinear Schrodinger equation for superfluids and Bose-Einstein condensate, water waves and their model equations (KDV, BBM, KP, Boussinesq systems...), Euler equations for inviscid fluids, ideal MHD for plasmas in fusion devices, Vlasov models for collisionless plasmas and galaxies, Yang-Mills equation in gauge field theory etc. There exist coherent structures (solitons, steady states, traveling waves, standing waves etc) which play an important role on the long time dynamics of these models. First, I will describe a general framework to study linear stability (instability) when the energy functional is bounded from below. For the models with indefinite energy functional (such as full water waves), approaches to find instability criteria will be mentioned. The implication of linear instability (stability) for nonlinear dynamics will be also briefly discussed.

In this talk, we will discuss what entails being a front-office quant at JP Morgan in the Emerging Markets group. We discuss why Emerging Markets is viewed as its own asset class and what there is to model. We also give practical examples of things we look at on a daily basis. This talk aims to be informal and to appeal to a wide audience.

There is a beautiful idea that one can study spaces by studying associated geometric objects. More specifically one can associate to a manifold (that is some space) a symplectic or contact manifold (that is the geometric object). The question is how useful is this idea. We will discuss this idea and related questions for subspaces (that is immersions and embeddings) with a focus on curves in the plane and knots in three space. If time permits we will discuss powerful new tools from contact geometry that allow one use this idea to construct invariants of knots and more generally embeddings and immersions in any space.