Filling metric spaces

Alexander Nabutovsky
University of Toronto; Member, School of Mathematics
March 8, 2019

Abstract:  Uryson k-width of a metric space X measures how close X is to being k-dimensional. Several years ago Larry Guth proved that if M is a closed n-dimensional manifold, and the volume of each ball of radius 1 in M does not exceed a certain small

constant e(n), then the Uryson (n-1)-width of M is less than 1. This result is a significant generalization of the famous Gromov's inequality relating the volume and the filling radius that plays a central role in systolic geometry.

 

Normalized harmonic map flow

Michael Struwe
ETH Zürich
March 6, 2019

Abstract: Finding non-constant harmonic 3-spheres for a closed target manifold N is a prototype of a super-critical variational problem. In fact, the 
direct method fails, as the infimum of the Dirichlet energy in any homotopy class of maps from the 3-sphere to any closed N is zero; moreover, the 
harmonic map heat flow may blow up in finite time, and even the identity map from the 3-sphere to itself is not stable under this flow. 

Existence and uniqueness of Green's function to a nonlinear Yamabe problem

Yanyan Li
Rutgers University
March 6, 2019

Abstract: For a given finite subset S of a compact Riemannian manifold (M; g) whose Schouten curvature tensor belongs to a given cone, we establish a necessary and
sufficient condition for the existence and uniqueness of a conformal metric on $M \setminus S$ such that each point of S corresponds to an asymptotically flat end and
that the Schouten tensor of the new conformal metric belongs to the boundary of the given cone.  This is a joint work with Luc Nguyen.

Rellich Kondrachov Theorem for L^2 curvatures in arbitrary dimension- Tristan Rivière

Tristan Rivière
ETH Zürich; Member, School of Mathematics
March 5, 2019

Abstract : What are the possible limits of smooth curvatures with uniformly bounded $L^p$ norms ?

We shall see that the attempts to give a satisfying answer to this natural question from the calculus of variation of gauge theory brings us to numerous analysis challenges.

Improved List-Decoding and Local List-Decoding Algorithms for Polynomial Codes

Swastik Kopparty
Rutgers University; Member, School of Mathematics
March 5, 2019

I will talk about a recent result showing that some well-studied polynomial-based error-correcting codes
(Folded Reed-Solomon Codes and Multiplicity Codes) are "list-decodable upto capacity with constant
list-size". 

At its core, this is a statement about questions of the form: "Given some points in the plane,
how many low degree univariate polynomials are such that their graphs pass through 10% of these points"? 

This leads to list-decodable and locally list-decodable error-correcting codes with the best known parameters.