Georgia Institute of Technology College of Sciences

It is well known that typically equations do not have analytic (expressed by formulas) solutions. Therefore a classical approach to the analysis of dynamical systems (from abstract areas of Math, e.g. the Number theory to Applied Math.) is to study their asymptotic (when an independent variable, "time", tends to infinity) behavior. Recently, quite surprisingly, it was demonstrated a possibility to study rigorously (at least some) interesting finite time properties of dynamical systems. Most of already obtained results are surprising, although rigorously proven. Possible PhD topics range from understanding these (already proven!) surprises and finding (and proving) new ones to numerical investigation of some systems/models in various areas of Math and applications, notably for dynamical analysis of dynamical networks. I'll present some visual examples, formulate some results and explain them (when I know how).

In his thesis, Margulis computed the asymptotic growth rate for the number of closed geodesics of length less than R on a given closed hyperbolic surface and his argument has been emulated to many other settings. We examine the Teichmüller geodesic flow on the moduli space of a surface, or more generally any stratum of quadratic differentials in the cotangent bundle of moduli space. The flow is known to be mixing, but the spaces are not compact and the flow is not uniformly hyperbolic. We show that the random walk associated to the Teichmüller geodesic flow is biased toward the compact part of the stratum. We then use this to find asymptotic growth rate of for the number of closed loops in the stratum. (This is a joint work with Alex Eskin and Maryam Mirzakhani.)

Waterborne diseases cause over 3.5 million deaths annually, with cholera alone responsible for 3-5 million cases/year and over 100,000 deaths/year. Many waterborne diseases exhibit multiple characteristic timescales or pathways of infection, which can be modeled as direct and indirect transmission. A major public health issue for waterborne diseases involves understanding the modes of transmission in order to improve control and prevention strategies. One question of interest is: given data for an outbreak, can we determine the role and relative importance of direct vs. environmental/waterborne routes of transmission? We examine these issues by exploring the identifiability and parameter estimation of a differential equation model of waterborne disease transmission dynamics. We use a novel differential algebra approach together with several numerical approaches to examine the theoretical and practical identifiability of a waterborne disease model and establish if it is possible to determine the transmission rates from outbreak case data (i.e. whether the transmission rates are identifiable). Our results show that both direct and environmental transmission routes are identifiable, though they become practically unidentifiable with fast water dynamics. Adding measurements of pathogen shedding or water concentration can improve identifiability and allow more accurate estimation of waterborne transmission parameters, as well as the basic reproduction number. Parameter estimation for a recent outbreak in Angola suggests that both transmission routes are needed to explain the observed cholera dynamics. I will also discuss some ongoing applications to the current cholera outbreak in Haiti.