Some interesting topics

Discrete charts, crystal defects and discrete curvature

Following Heitman-Radin's work which showed the crystallization of "sticky disk" models form a triangular lattice structure, there have been several developments towards a mathematical study of ground states with crystalline structure for discrete configurations. Modern approaches find robust connections to elasticity.

For example, if the configurations are not far from the reference perfect configurations, it is possible to encode the deformations as the gradients of a locally affine map. The existence of a reference configuration breaks down in the presence of too many defects or if we are in a curved ambient space for which a clear-cut reference configuration is not specified. In these situations one needs to understand geometrically the defects, and introduce notions of discrete curvature adapted to the interactions that we study.

This project consists in developing further the view that discrete configurations can be to a huge extent crystal-like even in curved situations, by extending notions of charts, parallel transport, curvature, and discrete versions of differential geometric invariants. After forming the geometric framework, these notions can then be ad-hoc modified for studying concrete minimization problems and energy behavior of almost crystalline point configurations with many defects.

Don't hesitate to write me if you are interested in these topics.

Stochastic geometry and integral geometry

The problems appearing in this field concern the "throwing at random" of geometric objects such as balls, the concept of a "random closed set" (i.e. what nice probability measures can be defined on closed sets, as opposed to measures on points, on a space?), or the average slicing of bodies by subspaces, curves, surfaces, etc.

The definition of Choquet capacities is also a part of the above field.

In a recent paper of mine with Codina Cotar (see paper A4 in the "papers section") we use a special type of random packings to localize the interaction in multimarginal optimal transport. It turns out that cutting off by a random packing is simply better than cutting off as usual.

This project would consist of learning more about stochastic geometry and applying this knowledge for Multimarginal Optimal Transport and other optimization problems. The theory of capacities also enters in a different part in paper A6 in my "papers" list.

Currently I am working on this theme with Daniel Pons from Universidad Andres Bello.

Don't hesitate to write me if you are interested in these topics.

Quantitative implicit function theorem and unique continuation

Starting from Weierstrass' and Malgrange's preparation theorems, up to the Nash-Moser implicit function theorem (see Wikipedia to know what they are!), there has been a strong mathematical tradition to understand generalizations and quantitative versions of implicit function theorems in Geometry, Analysis and Applications.

This is one of the fundamental basic tools for linking properties of functions and manifolds.

I am interested in developing further quantitative versions of the implicit function theorem, which can allow to prove, similar to the analytic functions case, unique continuation principles.

Don't hesitate to write me if you are interested in these topics.