Abstract: The nodal set of a Laplacian eigenfunction forms a partition of the underlying manifold. An alternative partition, based on the gradient field of the eigenfunction, is via the so called Neumann domains. A Neumann domain of an eigenfunction is a connected component of the intersection between the stable manifold of a certain minimum and the unstable manifold of a certain maximum. We introduce this subject, discuss various properties of Neumann domains and point out the similarities and differences between nodal domains and Neumann domains.

By a theorem of Borel and Harish-Chandra, an arithmetic group in a semisimple Lie group is a lattice. Conversely, by a celebrated theorem of Margulis, in a higher rank semisimple Lie group G any irreducible lattice is an arithmetic group. The aim of this lecture is to survey an arithmeticity criterium for discrete subgroups which are not assumed to be lattices. This criterium, obtained with Miquel, generalizes works of Selberg and Hee Oh and solves a conjecture of Margulis. It says: a discrete irreducible Zariski-dense subgroup of G that intersects cocompactly at least one

Abstract:

Abstract: Mirror symmetry is a correspondence between symplectic geometry on a manifold M and complex geometry on a mirror manifold W. The question of why one sort of geometry on M should be reflected in another sort of geometry on the topologically distinct manifold W, and the question of how to find W given M, are a priori highly mysterious. One attempt to explain the mysteries of mirror symmetry is the SYZ conjecture, which asserts that the mirror manifold W can be realized as the moduli space of certain objects of a category associated to M.