Mathematics Dept.
  Events
McGill
Tue 24 Oct 2017, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012
An isometric embedding problem and related geometric inequalities
ESB 2012
Tue 24 Oct 2017, 3:30pm-4:30pm

Abstract

Solutions to the classical Weyl problem by Nirenberg and Pogorelov
play fundamental role in the notion of quasi local masses and positive quasi
local mass theorems in general relativity. An interesting question in
differential geometry is whether one can isometrically embed compact surfaces
with positive Gauss curvature to a general 3 dimensional ambient space. Of
particular importance is the anti de Sitter Schwarzchild space in general
relativity.  We discuss some recent progress in this direction, the a priori
estimates for embedded surfaces in a joint work with Lu, the openness and
non-rigidity results of Li -Wang, and a new quasi local type inequality of
Lu-Miao. We will also discuss open problem related to isometric embeddings to
ambient spaces with horizons.
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Aaron Palmer
UBC
Tue 7 Nov 2017, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012
Optimal Stopping with a Probabilistic Constraint
ESB 2012
Tue 7 Nov 2017, 3:30pm-4:30pm

Abstract

Optimal stopping problems can be viewed as a problem to calculate
the space and time dependent value function, which solves a nonlinear, possibly
non-smooth and degenerate, parabolic PDE known as an Hamilton-Jacobi-Bellman
(HJB) equation.  These equations are well understood using the theory of
viscosity solutions, and the optimal stopping policy can be retrieved when
there is some regularity and non-degeneracy of solution.
 
The HJB equation is commonly derived from a dynamic programming principle
(DPP). After adding a probabilistic constraint, the optimal policies no longer
satisfy this DPP.  Instead, we can reach the HJB equation by a method related
to optimal transportation, and  recover a DPP for a Lagrangian-relaxation of
the problem.  The resulting HJB equation remains coupled through the constraint
with the optimal policy (and another parabolic PDE).  Solving the HJB and
recovery of the optimal stopping policy is aided by considering the
``piecewise-monotonic’' structure of the stopping set.
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Michal Kowalczyk
University of Chile
Tue 21 Nov 2017, 3:30pm
Diff. Geom, Math. Phys., PDE Seminar
ESB 2012
Asymptotic stability for some nonlinear Klein-Gordon equations for odd perturbations in the energy space
ESB 2012
Tue 21 Nov 2017, 3:30pm-4:30pm

Abstract

 Showing asymptotic stability in one dimensional nonlinear Klein-Gordon equations is a notoriously difficult problem. In this talk I will describe an approach based on virial estimates which allows to prove it in case when only odd perturbations are allowed. In particular I will discuss asymptotic stability of the kink in the $\phi^4$ model.       
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