There exist many different algorithms to compute the (approximate) inverse of an operator K to recover an approximation of a function f from data that represent Kf corrupted by noise. Recently, several approaches have been proposed that are adapted to the case when f has a sparse expansion in e.g. a wavelet basis. It turns out that one can find such an approximation to f by means of an iterative algorithm that uses repeatedly the simple thresholding operator that solves the problem when K = Id. The successive approximations converge in norm, and provide a stable regularization of the problem when the inverse problem is ill-conditioned. (This is joint work with Christine De Mol and Michel Defrise.)

The goals of this lecture are three-fold:

To discuss several operator-splitting methods (OSM) for the time-discretization of initial value problems.

To combine OSM with finite element and fictitious domain methods in order to simulate particulate flow for various types of incompressible viscous fluids.

To combine OSM with mixed finite element methods in order to solve highly nonlinear partial differential equations originating from differential geometry (and other fields) such as Eikonal, Monge-Ampè re's, Pucci's, etc. (see figure below).

This presentation will include the results of numerical experiments, validating the methodology under consideration. They show, in particular, the robustness, flexibility, and versatility of operator splitting methods.

Geometric evolution equations like the Ricciflow of metrics and the mean curvature flow of hypersurfaces can be used to smoothen and to uniformize the geometry of manifolds just like a heat equation can be used to approximate equilibrium states. In recent years and months possible singularities of Ricciflow and mean curvature flow have been understood in such detail that an extension past singularities has become possible leading to a classification of large classes of manifolds. The lecture describes joint work with C. Sinestrari on surgery procedures and longtime existence results for mean curvature flow of hypersurfaces. The methods in particular provide a complete classification of 3-dimensional hypersurfaces of positive scalar curvature in Euclidean 4-space. The lecture will illustrate major techniques and explain the relation of these results to the work of Hamilton and Perelmann on Riemannian 3-manifolds.

Vertex operator algebra theory is an inherently "non-classical" subject deeply related to "monstrous moonshine" and many other themes in mathematics, and to string theory in physics. I will motivate, introduce and sketch a selection of the main themes and problems, including some compelling current ones, in this exciting area.

The writer of puzzles often invents puzzles to illustrate a principle. The puzzles, however, sometimes have other ideas. Sometimes, they speak up and say that they would be so much prettier as slight variants of their original selves. The dilemma is that the puzzle-writer sometimes can't solve those variants. Sometimes he finds out that his colleagues can't solve them either, because there is no existing theory for solving them. We discuss a few such upstarts inspired originally from architecture, zero-knowledge proofs, diplomacy, prime numbers, and computational geometry. They have given a good deal of trouble to a certain mathematical detective whom I know well.