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 Events
UBC
Tue 4 Sep 2018, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
MATH 105 [The NEW LOCATION!]
The Dyson Game (joint work with R. Carmona and M. Cerenzia)
MATH 105 [The NEW LOCATION!]
Tue 4 Sep 2018, 3:30pm-4:30pm

Abstract

 The Dyson game is an explicitly solvable N player dynamic game that admits Dyson Brownian motion as a Nash equilibrium.  The game is motivated by the real world phenomenon found in the spacing of buses, parked cars and perched birds, which exhibit random matrix statistics (i.e. Dyson Brownian motion). We find the optimal repulsion parameter (universality class) of the equilibrium depends on the information available to the players, furthering the understanding of an open problem in random matrix theory proposed by Deift. The limiting mean field game has a local cost term, which depends on the optimal universality class due to the nontrivial asymptotic behavior of the players.  We solve the mean field game master equation and the associated Hamilton-Jacobi equation on Wasserstein space exactly, and we discuss how generalizing our results will require answering novel questions on the analysis of these equations on infinite dimensional spaces.
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Christos Mantoulidis
MIT
Tue 9 Oct 2018, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
MATH 105
Minimal surfaces and the Allen-Cahn equation on 3 manifolds
MATH 105
Tue 9 Oct 2018, 3:30pm-4:30pm

Abstract

   The Allen--Cahn equation is a semi-linear PDE that produces minimal surfaces via a certain singular limit. We will describe recent work proving index, multiplicity, and curvature estimates in the context of an Allen--Cahn min-max construction in a 3-manifold. Our results imply, for example, that in a 3-manifold with a generic metric, for every positive integer p, there is an embedded two-sided minimal surface of Morse index p. This is joint with Otis Chodosh.
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