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

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. 

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.

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. 


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. 

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: 

Kaehler constant scalar curvature metrics on blow ups and resolutions of singularities

Claudio Arezzo
International Centre for Theoretical Physics, Trieste
March 4, 2019

Abstract: After recalling the gluing construction for Kaehler constant scalar curvature and extremal (`a la Calabi) metrics
starting from a compact or ALE orbifolds with isolated singularities, I will show how to compute the Futaki invariant
of the adiabatic classes in this setting, extending previous work by Stoppa, Szekelyhidi and Odaka. Besides giving
new existence and non-existence results, the connection with the Tian-Yau-Donaldson Conjecture and the K-stability

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.