Georgia Institute of Technology College of Sciences

We consider the restricted planar elliptic 3 body problem, which models the Sun, Jupiter and an Asteroid (which we assume that has negligible mass). We take a realistic value of the mass ratio between Jupiter and the Sun and their eccentricity arbitrarily small and we study the regime of the mean motion resonance 1:7, namely when the period of the Asteroid is approximately seven times the period of Jupiter. It is well known that if one neglects the influence of Jupiter on the Asteroid, the orbit of the latter is an ellipse. In this talk we will show how the influence of Jupiter may cause a substantial change on the shape of Asteriod's orbit. This instability mechanism may give an explanation of the existence of the Kirkwood gaps in the Asteroid belt. This is a joint work with J. Fejoz, V. Kaloshin and P. Roldan.

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.

For two polynomials G(X), H(Y) with rational coefficients, when does G(X) = H(Y) have infinitely many solutions over the rationals? Such G and H have been classified in various special cases by previous mathematicians. A theorem of Faltings (the Mordell conjecture) states that we need only analyze curves with genus at most 1.In my thesis (and more recent work), I classify G(X) = H(Y) defining irreducible genus zero curves. In this talk I'll present the infinite families which arise in this classification, and discuss the techniques used to complete the classification.I will also discuss in some detail the examples of polynomial which occur in the classification. The most interesting infinite family of polynomials are those H(Y) solving a Pell Equation H(Y)^2 - P(Y)Q(Y)^2 = 1. It turns out to be difficult to describe these polynomials more explicitly, and yet we can completely analyze their decompositions, how many such polynomials there are of a fixed degree, which of them are defined over the rationals (as opposed to a larger field), and other properties.

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.)

A small set expander is a graph where every set of sufficiently small size has near perfect edge expansion. This talk concerns the computational problem of distinguishing a small set-expander, from a graph containing a small non-expanding set of vertices. This problem henceforth referred to as the Small-Set Expansion problem has proven to be intimately connected to the complexity of large classes of combinatorial optimization problems. More precisely, the small set expansion problem can be shown to be directly related to the well-known Unique Games Conjecture -- a conjecture that has numerous implications in approximation algorithms. In this talk, we motivate the problem, and survey recent work consisting of algorithms and interesting connections within graph expansion, and its relation to Unique Games Conjecture.