# School of Mathematics

## Loop products, closed geodesics and self-intersections

## Normalized harmonic map flow

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

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

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

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

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

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

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

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.