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.