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).
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.
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.
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.
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.
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.