School of Mathematics

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. 

Spacetime positive mass theorem

Lan-Hsuan Huang
University of Connecticut; von Neumann Fellow, School of Mathematics
March 5, 2019

Abstract: The spacetime positive mass theorem says that an asymptotically flat initial data set with the dominant energy condition must have a timelike energy-momentum vector, unless the initial data set is in the Minkowski spacetime. We will review backgrounds and recent progress toward this statement. 

Liouville Equations and Functional Determinants

Andrea Malchiodo
Scuola Normale Superiore
March 5, 2019

Abstract:  Functional Determinants are quantities constructed out of spectra of conformally covariant operators, and are explicit in dimension two and four, due to formulas by Polyakov and Branson-Oersted. Extremizing them in a conformal class amounts to solving Liouville equations with principal parts of different order but all scaling invariant. We discuss some existence, uniqueness, non-uniqueness results and some open problems. This is joint work with M.Gursky and P.Esposito. 

 

Periodic Geodesics and Geodesic Nets on Riemannian Manifolds

Regina Rotman
University of Toronto; Member, School of Mathematics
March 5, 2019

Abstract: I will talk about periodic geodesics, geodesic loops, and geodesic nets on Riemannian manifolds. More specifically, I will discuss some curvature-free upper bounds for compact manifolds and the existence results for non-compact manifolds. In particular, geodesic nets turn out to be useful for proving results about geodesic loops and periodic geodesics.

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. 

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.

Local and global expansion of graphs

Yuval Peled
New York University
March 4, 2019

The emerging theory of High-Dimensional Expansion suggests a number of inherently different notions to quantify expansion of simplicial complexes. We will talk about the notion of local spectral expansion, that plays a key role in recent advances in PCP theory, coding theory and counting complexity. Our focus is on bounded-degree complexes, where the problems can be stated in a graph-theoretic language: 

Gysin sequences and cohomology ring of symplectic fillings

Zhengyi Zhou
Member, School of Mathematics
March 4, 2019

It is conjectured that contact manifolds admitting flexible fillings have unique exact fillings. In this talk, I will show that exact fillings (with vanishing first Chern class) of a flexibly fillable contact (2n-1)-manifold share the same product structure on cohomology if one of the multipliers is of even degree smaller than n-1. The main argument uses Gysin sequences from symplectic cohomology twisted by sphere bundles.