Georgia Institute of Technology College of Sciences

Rare events, metastability and Monte Carlo methods for stochastic dynamical systems have been of central scientific interest for many years now. In this talk we focus on multiscale systems that can exhibit metastable behavior, such as rough energy landscapes. We discuss quenched large deviations in related random rough environments and design of provably efficient Monte Carlo methods, such as importance sampling, in order to estimate probabilities of rare events. Depending on the type of interaction of the fast scales with the strength of the noise we get different behavior, both for the large deviations and for the corresponding Monte Carlo methods. Standard Monte Carlo methods perform poorly in these kind of problems in the small noise limit. In the presence of multiple scales one faces additional difficulties and straightforward adaptation of importance sampling schemes for standard small noise diffusions will not produce efficient schemes. We resolve this issue and demonstrate the theoretical results by examples and simulation studies.

I will explain the construction of the essential skeleton of a one-parameter degeneration of algebraic varieties, which is a simplicial space encoding the geometry of the degeneration, and I will prove that it coincides with the skeleton of a good minimal dlt-model of the degeneration if the relative canonical sheaf is semi-ample. These results, contained in joint work with Mircea Mustata and Chenyang Xu, provide some interesting connections between Berkovich geometry and the Minimal Model Program.

In this talk, we will discuss a result due to Gabai which states that a minimal genus Seifert surface for a knot in 3-sphere can be realized as a leaf of a taut foliation of the knot complement. We will give a fairly detailed outline of the proof. In the process, we will learn how to construct taut foliations on knot complements.

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 also explore the evolution of droplet-like configurations (e.g., an infected region embedded in an all healthy state). 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.