# SCIENTIFIC PROGRAMS AND ACTIVITIES

February 21, 2018

THE FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES

July-December 2014
Thematic Program on
Variational Problems in Physics, Economics and Geometry

 Fall Semester Members Seminar Tuesdays 2:10 p.m. in Fields Room 230 Organizing Committee Young-Heon Kim (University of British Columbia) Robert McCann (University of Toronto) Mircea Petrache (Université Pierre et Marie Curie) Fall Semester Postdoctoral Seminar Tuesdays 1:10 p.m. Fields Room 210 Organizing Committee Jun Kitagawa (University of Toronto / The Fields Institute) Ihsan Topaloglu (McMaster University)

 Seminars Fall Semester Postdoctoral Seminar Andres Contreras (The Fields Institute) Local minimizers in Ginzburg-Landau theory We present results concerning the existence of stable vortex configurations in 2D and 3D Ginzburg-Landau. These results represent joint works with R.L. Jerrard and S. Serfaty. Tuesday. December 2, 2014 1:10 p.m. Room 210 Fall Semester Members Seminar Young-Heon Kim (University of British Columbia) Multimarginal optimal transport We explain some recent progress on multi-marginal optimal transport, where a family of mass distributions are matched in an optimal way. This is based on joint work with Brendan Pass. Tuesday, December 2, 2014 2:10 p.m. Room 230 Fall Semester Postdoctoral Seminar Fedor Soloviev (The Fields Institute) Integrability of pentagram maps and Lax representations (Slides) We discuss integrability of higher dimensional pentagram maps. These maps provide an explicit example when a discrete map "jumps" between different invariant tori leading to a generalized version of Arnold-Liouville theorem. This is a joint work with Boris Khesin. Wednesday. December 3, 2014 2:10 p.m. Room 210 Fall Semester Postdoctoral Seminar Jun Kitagawa (The Fields Institute / University of Toronto) The Aleksandrov estimate and its variants in Monge-Ampère equations The Aleksandrov estimate plays a central role in the regularity theory of weak solutions of the Monge-Amp{`\e}re equation, which was pioneered by Caffarelli in the early 90's. Rather than talk about the regularity theory itself, I will focus on this one estimate and its variants, for example which arise in regularity of the optimal transport problem. I will give some elementary proofs and talk about the geometric intuition in connection to convex geometry that lies behind this somewhat mysterious looking estimate. Time permitting, I will also talk about an Aleksandrov type estimate applicable to a new class of equations, which includes problems in geometric optics that are not optimal transport problems (joint work in progress with Nestor Guillen). November 4, 2014 1:10 p.m. Room 210 Analysis & Applied Math Seminar Slim Ibrahim (University of Victoria) Asymptotic derivation of the classical Magneto-Hydro-Dynamic system from Navier-Stokes-Maxwell The incompressible Magneto-Hydro-Dynamic (MHD) system is a classical and fundamental model in plasma physics. Although well known, its derivation from Navier-Stokes type equations has been so far formal. In this talk and after reviewing the results about the well-posedness, I show how an asymptotic analysis of such equations can rigorously lead to a such a derivation. The key points is a precise study of the weak stability in the Lorentz. This is a joint work with D. Arsenio (Paris 7) & N. Masmoudi (Courant) Friday, October 31, 2014 1:10 p.m. BA6183, Bahen Center, 40 St. George St. Fall Semester Postdoctoral Seminar Cyril Joel Batkam (The Fields Institute) A symmetric mountain pass theorem for strongly indefinite functionals The symmetric mountain pass theorem of Ambrosetti and Rabinowitz (1973) and its generalization by Bartsch (1993) are effective tools of finding high energy solutions to many (partial) differential equations and systems which are of variational nature and exhibit some symmetry properties. A functional defined on a Hilbert space fits into the framework of these critical point theorems only if its quadratic part has a finite number of negative eigenvalues. In this talk, we present a generalization of these results to the case where the quadratic part has infinitely many negative eigenvalues (strongly indefinite functional). As an application, we give a direct proof of the existence of infinitely many solutions to the system \begin{equation*} \left\{ \begin{array}{ll} -\Delta u=g(x,v)\,\, \text{in }\Omega, & \hbox{} \\ -\Delta v=f(x,u)\,\,\text{in }\Omega, & \hbox{} \\ u=v=0\text{ on }\partial\Omega, & \hbox{} \end{array} \right. \end{equation*} where $\Omega$ is a bounded smooth domain in $\mathbb{R}^N$ ($N\geq3$), $f(x,u)\backsim |u|^{p-2}u$ and $g(x,v)\backsim |v|^{q-2}v$, with \$2