## Colloquium-Jeremy Tyson (UIUC)-Sobolev spaces, Lipschitz homotopy groups and the Heisenberg group

Mathematics - Colloquium

Friday, November 8, 2013

11:00 AM-12:00 PM

Stratton Hall

203

ABSTRACT: The theory of Sobolev spaces quantifies intermediate degrees of weak regularity for functions and mappings acting between geometric spaces.Originally formulated in the smooth category (e.g., on Euclidean spaces or on smooth Riemannian manifolds), Sobolev space theory has recently been extended to a wide class of metric measure spaces. Motivations for such developments come from partial differential equations (geometric analysis of general hypoelliptic and subelliptic operators arising in several complex variables, control theory, etc.), geometric function theory(quasiconformal mappings on non-Riemannian spaces, with applications to rigidity problems for negatively curved manifolds), analysis on fractals, and many other areas.

In this talk I will discuss the problem of density of smooth or Lipschitz mappings in Sobolev spaces. This problem is classical for Euclidean targets, semi-classical for Riemannian targets, and a topic of current research interest for general metric space targets. I will describe recent results on the problem of density of Lipschitz maps in Sobolev spaces of mappings valued in the sub-Riemannian Heisenberg group. The solution to this problem involves some novel ingredients arising from Lipschitz algebraic topology.

Suggested Audiences:
Adult, College

E-mail:
ma-chair@wpi.edu

Last Modified: October 25, 2013 at 9:19 AM

## Colloquium-Jeremy Tyson (UIUC)-Sobolev spaces, Lipschitz homotopy groups and the Heisenberg group

Mathematics - Colloquium

Friday, November 8, 2013

11:00 AM-12:00 PM

Stratton Hall

203

ABSTRACT: The theory of Sobolev spaces quantifies intermediate degrees of weak regularity for functions and mappings acting between geometric spaces.Originally formulated in the smooth category (e.g., on Euclidean spaces or on smooth Riemannian manifolds), Sobolev space theory has recently been extended to a wide class of metric measure spaces. Motivations for such developments come from partial differential equations (geometric analysis of general hypoelliptic and subelliptic operators arising in several complex variables, control theory, etc.), geometric function theory(quasiconformal mappings on non-Riemannian spaces, with applications to rigidity problems for negatively curved manifolds), analysis on fractals, and many other areas.

In this talk I will discuss the problem of density of smooth or Lipschitz mappings in Sobolev spaces. This problem is classical for Euclidean targets, semi-classical for Riemannian targets, and a topic of current research interest for general metric space targets. I will describe recent results on the problem of density of Lipschitz maps in Sobolev spaces of mappings valued in the sub-Riemannian Heisenberg group. The solution to this problem involves some novel ingredients arising from Lipschitz algebraic topology.

Suggested Audiences: Adult, College

E-mail: ma-chair@wpi.edu

Last Modified: October 25, 2013 at 9:19 AM