# School of Mathematics

## $L^2$ curvature for surfaces in Riemannian manifolds

Abstract: For surfaces immersed into a compact Riemannian manifold, we consider the curvature functional given by the $L^2$ integral of the second fundamental form. We discuss an an area bound in terms of that functional, with application to the existence of minimizers (joint work with V. Bangert).

## 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.

## Loop products, closed geodesics and self-intersections

## Nature of some stationary varifolds near multiplicity 2 tangent planes

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

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

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

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.

## 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.

## 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.