I will consider classical 2D traveling water waves with vorticity. By
means of local and global bifurcation theory using topological degree,
we can prove that there exist many such waves. They are exact smooth
solutions of the Euler equations with the physical boundary conditions.
Some of the waves are quite tall and steep and some are overhanging.
There are periodic ones and solitary ones. I will exhibit some
numerical computations of such waves. New analytical results will be
presented on waves with favorable vorticity.
Title: Counting points and counting representations
I will talk about the following questions:
1) Given a system of polynomial equations with integer coefficients, how many solutions does it have in the ring Z/N?
2) Given a polynomial map f:R^a-->R^b and a smooth, compactly supported measure m on R^a, does the push-forward of m by f have bounded density?
3) Given a lattice in a higher rank Lie group (say, SL(n,Z) for n>2). How many d-dimensional representations does it have?
Coherent configurations" (CCs) are certain highly regular colorings of the directed complete graph. The concept goes back to Schur (1933) who used it to study permutation groups, and has subsequently been rediscovered in other contexts (block designs,
association schemes, graph canonization).
CCs are the central concept in the "Split-or-Johnson" (SoJ) procedure, one of the main combinatorial components of the speaker's recent algorithm to test graph isomorphism.
Title: Avatars of small cancellation
In general, given a finite presentation of a group, it is very difficult (in fact algorithmically impossible) to understand the group it defines. Small cancellation theory was developped as a combinatorial condition on a presentation that allows one to understand the group it represents. This very flexible construction has many applications to construct examples of groups with specific features.