Georgia Institute of Technology College of Sciences

Many complex models from science and engineering can be studied in the framework of coupled systems of differential equations on networks. A network is given by a directed graph. A local system is defined on each vertex, and directed edges represent couplings among vertex systems. Questions such as stability in the large, synchronization, and complexity in terms of dynamic clusters are of interest. A more recent approach is to investigate the connections between network topology and dynamical behaviours. I will present some recent results on the construction of global Lyapunov functions for coupled systems on networks using a graph theoretic approach, and show how such a construction can help us to establish global behaviours of compelx models.

We will explain the concept of aproximate well-supported Nash equilibrium and show that one must consider equilibria with large supports to achieve good approximation ratio. Our arguments use tools from probabilistic, extremal and additive combinatorics. Joint work with Y. Anbalagan, R. Savani and A. Vetta.

We say that a cover of surfaces S-> X has the Birman--Hilden property if the subgroup of the mapping class group of X consisting of mapping classes that have representatives that lift to S embeds in the mapping class group of S modulo the group of deck transformations. We identify one necessary condition and one sufficient condition for when a cover has this property. We give new explicit examples of irregular branched covers that do not satisfy the necessary condition as well as explicit covers that satisfy the sufficient condition. Our criteria are conditions on simple closed curves, and our proofs use the combinatorial topology of curves on surfaces.

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.