Georgia Institute of Technology College of Sciences

Given any smooth manifold, there is a canonical symplectic structure on its cotangent bundle. A long standing idea of Arnol'd suggests that the symplectic topology of the cotangent bundle should contain a great deal of information about the smooth topology of its base. As a contrast, I show that when X is an open 4-manifold, this symplectic structure on T^*X does not depend on the choice of smooth structure on X. I will also discuss the particular cases of smooth structures on R^4 and once-punctured compact 4-manifolds.

We present a method for the detection of stable and unstable fibers of invariant manifolds of periodic orbits. We show how to propagate the fibers to prove transversal intersections of invariant manifolds. The method can be applied using interval arithmetic to produce rigorous, computer assisted estimates for the manifolds. We apply the method to prove transversal intersections of stable and unstable manifolds of Lyapunov orbits in the restricted three body problem.

(algebraic statistics reading seminar)

We study the Dirichlet and Neumann type initial-boundary value problems for strongly degenerate parabolic-hyperbolic equations. We suggest the notions of entropy solutions for these problems and establish the uniqueness of entropy solutions. The existence of entropy solutions is also discussed（joint work with Yuxi Hu and Qin Wang).

We will present the method of correctly aligned windows and show how it can lead to large scale motions when there are homoclinic orbits to a normally hyperbolic manifold.

For many evolutionary dynamics, within a population there are finitely many types that compete with each other. If we think of a type as a strategy, we may consider this dynamic from a game theoretic perspective. This evolution is frequency dependent, where the fitness of each type is given by the expected payoff for an individual in that subpopulation. Considering the frequencies of the population, the logarithmic growth is given by the difference of the respective fitness and the average fitness of the population as a whole. This dynamic is Darwinian in nature, where Nash Equilibria are fixed points, and Evolutionary Stable Strategies are asymptotically stable. Fudenberg and Harris modified this deterministic dynamic by assuming the fitness of each type are subject to population level shocks, which they model by Brownian motion. The authors characterize the two strategy case, while various other authors considered the arbitrary finite strategy case, as well as different variations of this model. Considering how ecological and social anomalies affect fitness, I expand upon the Fudenberg and Harris model by adding a compensated Poisson term. This type of stochastic differential equation is no longer continuous, which complicates the analysis of the model. We will discuss the approximation of the 2 strategy case, stability of Evolutionary Stable Strategies and extinction of dominated strategies for the arbitrary finite strategy case. Examples of applications are given. Prior knowledge of game theory is not needed for this talk.

After some brief comments about the nature of mathematical modeling in biology and medicine, we will formulate and analyze the SIR infectious disease transmission model. The model is a system of three non-linear differential equations that does not admit a closed form solution. However, we can apply methods of dynamical systems to understand a great deal about the nature of solutions. Along the way we will use this model to develop a theoretical foundation for public health interventions, and we will observe how the model yields several fundamental insights (e.g., threshold for infection, herd immunity, etc.) that could not be obtained any other way. At the end of the talk we will compare the model predictions with data from actual outbreaks.

TBA

This is an expository talk on the arc complex and translation distance of open book decompositions. We will discuss curve complexes, arc complex, open books, and finally the application to contact manifolds.

We will study one and two weight inequalities for several different operators from harmonic analysis, with an emphasis on vector-valued operators. A large portion of current research in the area of one weight inequalities is devoted to estimating a given operators' norm in terms of a weight's A_p characteristic; we consider some related problems and the extension of several results to the vector-valued setting. In the two weight setting we consider some of the difficulties of characterizing a two weight inequality through Sawyer-type testing conditions.

The periodic generalized Korteweg-de Vries equation (gKdV) is a canonical dispersive partial differential equation with numerous applications in physics and engineering. In this talk we present invariance of the Gibbs measure under the flow of the gauge transformed periodic quartic gKdV. The proof relies on probabilistic arguments which exhibit nonlinear smoothing when the initial data are randomized. As a corollary we obtain almost sure global well-posedness for the (ungauged) quartic gKdV at regularities where this PDE is deterministically ill-posed.

In this series of talks I will begin by discussing the idea of studying smooth manifolds and their submanifolds using the symplectic (and contact) geometry of their cotangent bundles. I will then discuss Legendrian contact homology, a powerful invariant of Legendrian submanifolds of contact manifolds. After discussing the theory of contact homology, examples and useful computational techniques, I will combine this with the conormal discussion to define Knot Contact Homology and discuss its many wonders properties and conjectures concerning its connection to other invariants of knots in S^3.

In this talk, we are going to introduce Linearized Proximal Alternating Minimization Algorithm and its variants for total variation based variational model. Since the proposed method does not require any special inner solver (e.g. FFT or DCT), which is quite often required in augmented Lagrangian based approach (ADMM), it shows better performance for large scale problems. In addition, we briefly introduce new regularization method (nonconvex higher order total variation).

A permutation of the set {1,2,...,n} is connected if there is no k < n such that the set of the first k numbers is invariant as a set under the permutation. For each permutation, there is a corresponding graph whose vertices are the letters of the permutation and whose edges correspond to the inversions in the permutation. In this way, connected permutations correspond to connected permutation graphs. We find a growth process of a random permutation in which we start with the identity permutation on a fixed set of letters and increase the number of inversions one at a time. After the m-th step of the process, we obtain a random permutation s(n,m) that is uniformly distributed over all permutations of {1,2,...,n} with m inversions. We will discuss the evolution process, the connectedness threshold for the number of inversions of s(n,m), and the sizes of the components when m is near the threshold value. This study fits into the wider framework of random graphs since it is analogous to studying phase transitions in random graphs. It is a joint work with my adviser Boris Pittel.

I'll discuss two methods for finding bounds on sums of graph eigenvalues (variously for the Laplacian, the renormalized Laplacian, or the adjacency matrix). One of these relies on a Chebyshev-type estimate of the statistics of a subsample of an ordered sequence, and the other is an adaptation of a variational argument used by P. Kröger for Neumann Laplacians. Some of the inequalities are sharp in suitable senses. This is ongoing work with J. Stubbe of EPFL