The de Rham complex is a prototype for a large class of
sequences of differential operators often called (generalized)
Bernstein-Gelfand-Gelfand BGG sequences. Conformal manifolds admit
such sequences and on locally conformally flat manifolds the...
In 1979, in collaboration with D. Bennequin, we started a
direction of research around the study of periodic trajectories of
the Reeb vector field $\xi$ on a contact manifold $(M^3, \alpha)$.
We will describe in this talk where this direction of...
We address the problem of building a body of specified shape and
of specified mass, out of materials of varying density so as to
minimize the first Dirichlet eigenvalue. It leads to a free
boundary problem and many uniqueness questions, The...
In optimal transport theory, one wants to understand the
phenomena arising when mass is transported in a cheapest way. This
variational problem is governed by the structure of the
transportation cost function defined on the product of the source
and...
A well-known example by N. N. Ural'tseva suggests that for fixed
p > 2 there is no unique $W^2_p$-solvability of elliptic
equations under p > the condition that the leading coefficients
are measurable in two spatial variables. We will present a...
In this talk, we shall review the convexity of solutions of
elliptic partial differential equations; we concentrate on the
constant rank theorem for the hessian of the convex solution. As
for the interesting from geometry problems, recently we have...
The special Lagrangian equations define calibrated minimal
Lagrangian surfaces in complex space. These fully nonlinear Hessian
equations can also be written in terms of symmetric polynomials of
the Hessian, giving a minimal surface interpretation to...
In these lectures we will describe the relationship between
optimal transportation and nonlinear elliptic PDE of Monge-Ampere
type, focusing on recent advances in characterizing costs and
domains for which the Monge-Kantorovich problem has smooth...
