Colloquium[ Edit ]
Moderator: Ron Rosenthal
For any element w in the free group on k generators and a group G, there is a map G^k --> G is defined by substitution. These so-called word maps are the analogs of polynomial maps in the category of groups. I will talk about the images of word maps in arithmetic groups, and then segue into the model theory of these groups ending with new rigidity phenomenon for them. The non-survey parts of this work are joint with Alex Lubotzky and Chen Meiri.
The nodal set of a nice function defined on a smooth manifold or the Euclidean space is its zero set. The study of nodal sets of Gaussian random fields has positioned itself in the heart of several disciplines, including probability theory and spectral geometry, and, more recently, it has exhibited connections to number theory. We are interested in the asymptotic topology and geometry of the nodal lines in the high energy limit.
In the first part of the talk I will give an overview of the classical results in this field, and the related methods. In the second part of the talk I will describe the more recent progress,related to percolation properties of the nodal lines, borrowing from percolation theory, inspired by the beautiful percolation model due to Bogomolny-Schmit. Finally, I will describethe recent results obtained in a joint work with D. Beliaev and S. Muirhead on the relation between the percolation properties of the nodal sets and their connectivity measures, that were defined and whose existence was established in a joint work with P. Sarnak.