Mon, 18 Mar 2019
14:15
C4

Invariants for sublinearly biLipschitz equivalence

Gabriel Pallier
(Université Paris-Sud 11)
Abstract


The large-scale features of groups and spaces are recorded by asymptotic invariants. Examples of asymptotic invariants are the asymptotic cone and, for hyperbolic groups, the Gromov boundary.
In his study of asymptotic cones of connected Lie groups, Yves Cornulier introduced a class of maps called sublinearly biLipschitz equivalences. Like the more traditionnal quasiisometries, sublinearly biLipschitz equivalences are biLipschitz on the large-scale, but unlike quasiisometries, they are generally not coarse. Sublinearly biLipschitz equivalences still induce biLipschitz homeomorphisms between asymptotic cones. In this talk, I will focus on Gromov-hyperbolic groups and show how the Gromov boundary can be used to produce invariants distinguishing them up to sublinearly biLipschitz equivalences when the asymptotic cones do not. I will especially give applications to the large-scale sublinear geometry of hyperbolic Lie groups.
 

Thu, 11 Mar 2010

14:30 - 15:30
Gibson 1st Floor SR

Entropy method for line-energies

Radu Ignat
(Université Paris-Sud 11)
Abstract

The aim of this talk is to analyze energy functionals concentrated on the jump set of 2D vector fields of unit length and of vanishing divergence.

The motivation of this study comes from thin-film micromagnetics where these functionals correspond to limiting wall-energies. The main issue consists in characterizing the wall-energy density (the cost function) so that the energy functional is lower semicontinuous (l.s.c.). The key point resides in the concept of entropies due to the scalar conservation law implied by our vector fields. Our main result identifies appropriate cost functions

associated to certain sets of entropies. In particular, certain power cost functions lead to l.s.c. energy functionals.

A second issue concerns the existence of minimizers of such energy functionals that we prove via a compactness result. A natural question is whether the viscosity solution is a minimizing configuration. We show that in general it is not the case for nonconvex domains.

However, the case of convex domains is still open. It is a joint work with Benoit Merlet, Ecole Polytechnique (Paris).

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