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 Events
University of Alberta
Thu 8 Dec 2016, 4:00pm SPECIAL
Diff. Geom, Math. Phys., PDE Seminar
ESB Room 4127 (PIMS Videoconferencing Room )
Mulit-to one-dimensional optimal transport
ESB Room 4127 (PIMS Videoconferencing Room )
Thu 8 Dec 2016, 4:00pm-5:00pm

Abstract

I will discuss joint work with Pierre-Andre Chiappori and Robert McCann on the Monge-Kantorovich problem of transporting a probability measure on \mathbb{R}^n to another on the real line. We introduce a nestededness criterion relating the cost to the marginals, under which it is possible to solve this problem uniquely (and essentially explicitly), by constructing an optimal map one level set at a time. I plan to discuss examples for which the nestedness condition holds, as well as some for which it fails; some of these examples arise from a matching problem in economics which originally motivated our work. If time permits, I will also briefly discuss how level set dynamics can be used to develop a local regularity theory in the nested case
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Ph.D. Candidate: Zichun Ye
Mathematics, UBC
Mon 12 Dec 2016, 12:30pm SPECIAL
Room 200, Graduate Student Centre
Doctoral Exam: Models of Gradient Type with Sub-Quadratic Actions and Their Scaling Limits
Room 200, Graduate Student Centre
Mon 12 Dec 2016, 12:30pm-2:30pm

Details

My research concerns models of gradient type with sub-quadratic actions and their scaling limits. The model of gradient type is the density of a collection of real-valued random variables ϕ’s given by Z^{-1}e^({-ΣV(ϕ_j-ϕ_k)}). We focus our study on the case that V(t) = [1+t^2]^a with 0 < a < 1/2, which is a non-convex potential.

The first result concerns the thermodynamic limits of the model of gradient type. We introduce an auxiliary field t for each edge and represent the model as the marginal of a model with log-concave density. Based on this method, we prove that finite moments of the fields are bounded uniformly in the volume for the finite volume measure. This bound leads to the existence of infinite volume measures.

The second result is the random walk representation and the scaling limit of the translation-invariant, ergodic gradient infinite volume Gibbs measure. We represent every infinite volume Gibbs measure as a mixture over Gaussian gradient measures with a random coupling constant ω for each edge. With such representation, we give estimations on the decay of the two point correlation function. Then by the quenched functional central limit theorem in random conductance model, we prove that every ergodic, infinite volume Gibbs measure with mean zero for the potential V above scales to a Gaussian free field.
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