Georgia Institute of Technology College of Sciences

This is a review talk on an infinite dimensional relaxation of mixed integer programs (MIP) that was developed by Gomory and Johnson. We will discuss the relationship between cutting planes for the original MIP and its infinite dimensional relaxation. Time permitting, various structural results about the infinite dimensional problem and some open problems will be presented.

The Kardar-Parisi-Zhang(KPZ) equation is a non-linear stochastic partial di fferential equation proposed as the scaling limit for random growth models in physics. This equation is, in standard terms, ill posed and the notion of solution has attracted considerable attention in recent years. The purpose of this talk is two fold; on one side, an introduction to the KPZ equation and the so called KPZ universality classes is given. On the other side, we give recent results that generalize the notion of viscosity solutions from deterministic PDE to the stochastic case and apply these results to the KPZ equation. The main technical tool for this program to go through is a non-linear version of Feyman-Kac's formula that uses Doubly Backward Stochastic Differential Equations (Stochastic Differential Equations with times flowing backwards and forwards at the same time) as a basis for the representation.

Fractional calculus (FC) provides a powerful formalism for the modeling of systems whose underlying dynamics is governed by Lévy stochastic processes. In this talk we focus on two applications of FC: (1) non-diffusive transport, and (2) option pricing in finance. Regarding (1), starting from the continuous time random walk model for general Lévy jump distribution functions with memory, we construct effective non-diffusive transport models for the spatiotemporal evolution of the probability density function of particle displacements in the long-wavelength, time-asymptotic limit. Of particular interest is the development of models in finite-size-domains and those incorporating tempered Lévy processes. For the second application, we discuss fractional models of option prices in markets with jumps. Financial instruments that derive their value from assets following FMLS, CGMY, and KoBoL Lévy processes satisfy fractional diffusion equations (FDEs). We discuss accurate, efficient methods for the numerical integration of these FDEs, and apply them to price barrier options. The numerical methods are based on the finite difference discretization of the regularized fractional derivatives in the Grunwald-Letnikov representation.

Abstract: Dynamical systems with a coupled cell network structure arise in applications that range from statistical mechanics and electrical circuits to neural networks, systems biology, power grids and the world wide web. A network structure can have a strong impact on the behaviour of a dynamical system. For example, it has been observed that networks can robustly exhibit (partial) synchronisation, multiple eigenvalues and degenerate bifurcations. In this talk I will explain how semigroups and their representations can be used to understand and predict these phenomena. As an application of our theory, I will discuss how a simple feed-forward motif can act as an amplifier. This is joint work with Jan Sanders.