Oxford

This is joint with with Mark Berman, Uri Onn, and Pirita Paajanen.

Let K be a local field with valuation ring O and residue field of size q, and G a Chevalley group. We study counting problems associated with the group G(O). Such counting problems are encoded in certain zeta functions defined as Poincare series in q^{-s}. It turns out that these zeta functions are bounded sums of rational functions and depend only on q for all local fields of sufficiently large residue characteristic. We apply this to zeta functions counting conjugacy classes or dimensions of Hecke modules of interwining operators in congruence quotients of G(O). To prove this we use model-theoretic cell decomposition and quantifier-elimination to get a theorem on the values of 'definable' integrals over local fields as the local field varies.

I will describe why e^{\pi\sqrt{163}} is almost an integer and how this is related to Q(\sqrt{-163}) having class number one and why n^2-n+41 is prime for n=0,...,39. Bits and pieces about Gauss's class number problem, Heegner numbers, the j-invariant and complex multiplication on elliptic curves will be discussed along the way.

We give an introduction to the Kakimizu complex of a link,

covering a number of recent results. In particular we will see that the

Kakimizu complex of a knot may be locally infinite, that the Alexander

polynomial of an alternating link carries information about its Seifert

surfaces, and that the Kakimizu complex of a special alternating link is

understood.