Seminars and Colloquia by Series

Monday, February 1, 2016 - 11:00 , Location: Skiles 005 , Geng Chen , Georgia Tech , Organizer:
The nonlinear wave equation: u_{tt} - c(u)[c(u)u_x]_x = 0 is a natural generalization of the linear wave equation. In this talk, we will discuss a recent breakthrough addressing the Lipschitz continuous dependence of solutions on initial data for this quasi-linear wave equation. Our earlier results showed that this equation determines a unique flow of conservative solution within the natural energy space H^1(R). However, this flow is not Lipschitz continuous with respect to the H^1 distance, due to the formation of singularity. To prove the desired Lipschitz continuous property, we constructed a new Finsler type metric, where the norm of tangent vectors is defined in terms of an optimal transportation problem. For paths of piecewise smooth solutions, we carefully estimated how the distance grows in time. To complete the construction, we proved that the family of piecewise smooth solutions is dense, following by an application of Thom's transversality theorem. This is a collaboration work with Alberto Bressan.
Monday, December 7, 2015 - 11:00 , Location: Skiles 005 , Cristel Chandre , Centre de Physique Theorique CNRS Campus de Luminy , Organizer: Rafael de la Llave
Solving numerically kinetic equations requires high computing power and storage capacity, which compels us to derive more tractable, dimensionally reduced models. Here we investigate fluid models derived from kinetic equations, typically the Vlasov equation. These models have a lower numerical cost and are usually more tangible than their kinetic counterpart as they describe the time evolution of quantities such as the density ρ, the fluid velocity u, the pressure p, etc. The reduction procedure naturally leads to the need for a closure of the resulting fluid equations, which can be based on various assumptions. We present here a strategy for building fluid models from kinetic equations while preserving their Hamiltonian structure. Joint work with M. Perin and E. Tassi (CNRS/Aix-Marseille University) and P.J. Morrison (University of Texas at Austin).
Friday, December 4, 2015 - 11:00 , Location: Skiles 005 , Ke Zhang , Univ. of Toronto , Organizer: Rafael de la Llave
The study of random Hamilton-Jacobi PDE is motivated by mathematical physics, and in particular, the study of random Burgers equations. We will show that, almost surely, there is a unique stationary solution, which also has better regularity than expected. The solution to any initial value problem converges to the stationary solution exponentially fast. These properties are closely related to the hyperbolicity of global minimizer for the underlying Lagrangian system. Our result generalizes the one-dimensional result of E, Khanin, Mazel and Sinai to arbitrary dimensions.  Based on joint works with K. Khanin and R. Iturriaga.
Friday, November 20, 2015 - 15:00 , Location: Skiles 006 , Yingfei Yi , University of Alberta & Georgia Tech , Organizer: Rafael de la Llave
The talk concerns limit behaviors of stationary measures of diffusion processes generated from white-noise perturbed systems of ordinary differential equations. By relaxing the notion of Lyapunov functions associated with the stationary Fokker-Planck equations, new existence and non-existence results of stationary measures will be presented. As noises vanish, concentration and limit behaviors of stationary measures will be described with particular attentions paying to the special role played by multiplicative noises. Connections to problems such as stochastic stability, stochastic bifurcations, and thermodynamics limits will also be discussed.
Monday, November 16, 2015 - 11:00 , Location: Skiles 005 , James Tanis , College de France , Organizer: Rafael de la Llave
We prove results concerning the equidistribution of some "sparse" subsets of orbits of horocycle flows on $SL(2, R)$ mod lattice. As a consequence of our analysis, we recover the best known rate of growth of Fourier coefficients of cusp forms for arbitrary noncompact lattices of $SL(2, R)$, up to a logarithmic factor.  This talk addresses joint work with Livio Flaminio, Giovanni Forni and Pankaj Vishe.
Friday, November 6, 2015 - 11:00 , Location: Skiles 005 , Jon Fickenscher , Princeton University , , Organizer: Michael Damron
We will consider (sub)shifts with complexity such that the difference from n to n+1 is constant for all large n. The shifts that arise naturally from interval exchange transformations belong to this class. An interval exchange transformation on d intervals has at most d/2 ergodic probability measures. We look to establish the correct bound for shifts with constant complexity growth. To this end, we give our current bound and discuss further improvements when more assumptions are allowed. This is ongoing work with Michael Damron.
Monday, November 2, 2015 - 11:00 , Location: Skiles 005 , Siddharth Maddali , Carnegie Mellon , Organizer: Rafael de la Llave
I present a formalism and an computational scheme to quantify the dynamics of grain boundary migration in polycrystalline materials, applicable to three-dimensional  microstructure data obtained from non-destructive coarsening experiments. I will describe a geometric technique of interface tracking using well-established optimization algorithms and demonstrate how, when coupled with very basic physical assumptions, one can effectively measure grain boundary energy density and mobility of a given misorientation type in the two-parameter subspace of boundary inclinations. By doing away with any specific model or parameterization for the energetics, I seek to have my analysis applicable to general anisotropies in energy and mobility. I present results in two proof-of-concept test cases, one first described in closed form by J. von Neumann more than half a century ago, and the other which assumes analytic but anisotropic energy and mobility known in advance.
Wednesday, October 14, 2015 - 11:00 , Location: Skiles 05 , Blaz Mramor , Univ. Freiburg , Organizer: Rafael de la Llave
  The Allen-Cahn equation is a second order semilinear elliptic PDE that arises in mathematical models describing phase transitions between two constant states. The variational structure of this equation allows us to study energy-minimal phase transitions, which correspond to uniformly bounded non-constant globally minimal solutions. The set of such solutions depends heavily on the geometry of the underlying space. In this talk we shall focus on the case where the underlying space is a Cayley graph of a group with the word metric. More precisely, we assume that the group is hyperbolic and show that there exists a minimal solution with any “nice enough” asymptotic behaviour prescribed by the two constant states. The set in the Cayley graph where the phase transition for such a solution takes place corresponds to a solution of an asymptotic Plateau problem.
Monday, September 28, 2015 - 11:00 , Location: Skiles 005 , Chenchen Mou , Georgia Institute of Technology , Organizer:
 In this talk, we will consider semiconcavity of viscosity solutions for a class of degenerate elliptic integro-differential equations in R^n. This class of equations includes Bellman equations containing operators of Levy-Ito type. Holder and Lipschitz continuity of viscosity solutions for a more general class of degenerate elliptic integro-differential equations are also provided.
Monday, September 21, 2015 - 11:00 , Location: Skiles 005 , Marian Gidea , Yeshiva University , Organizer: Rafael de la Llave
We prove the existence of diffusion orbits drifting along heteroclinic chains of normally hyperbolic 3-dimensional cylinders, under suitable assumptions on the dynamics on the cylinders and on their homoclinic/heteroclinic connections. These assumptions are satisfied in the a priori stable case of the Arnold diffusion problem. We provide  a geometric argument that extends Birkhoff's procedure for constructing connecting orbits  inside a zone of instability for a twist map on the annuls. This is joint work with J.-P. Marco.