L3

How many triangles must a graph of density d contain? This old question due to Erdos was recently answered by Razborov, after many decades of progress by numerous authors.

We will consider the analogous question for tripartite graphs. Given a tripartite graph with prescribed edges densities between each

pair of classes how many triangles must it contain?

Combinatorial pattern matching is a subject which has given us fast and elegant algorithms for a number of practical real world problems as well as being of great theoretical interest. However, when single character wildcards or so-called "don't know" symbols are introduced into the input, classic methods break down and it becomes much more challenging to find provably fast solutions. This talk will give a brief overview of recent results in the area of pattern matching with don't knows and show how techniques from fields as disperse FFTs, group testing and algebraic coding theory have been required to make any progress. We will, if time permits, also discuss the main open problems in the area.

We revisit the problem of sorting under partial information: sort a finite set given the outcomes of comparisons between some pairs of elements. The input is a partially ordered set P, and solving the problem amounts to discovering an unknown linear extension of P, using pairwise comparisons. The information-theoretic lower bound on the number of comparisons needed in the worst case is log e(P), the binary logarithm of the number of linear extensions of P. In a breakthrough paper, Jeff Kahn and Jeong Han Kim (STOC 1992) showed that there exists a polynomial-time sorting algorithm achieving this bound up to a constant factor. They established a crucial link between the entropy of the input partial order and the information-theoretic lower bound. However, their algorithm invokes the ellipsoid algorithm at each iteration for determining the next comparison, making it unpractical. We develop efficient algorithms for sorting under partial information, derived from approximation and exact algorithms for computing the partial order entropy.

This is joint work with S. Fiorini, G. Joret, R. Jungers, and I. Munro.

The problem of checking existence for an Euler tour of a graph is trivial (are all vertex degrees even?). The problem of counting (or even approximate counting) Euler tours seems to be very difficult. I will describe two simple classes of graphs where the problem can be

solved exactly in polynomial time. And also talk about the many many classes of graphs where no positive results are known.

Given a polynomial function f defined on a variety X,

we consider two questions, which are non-commutative analogues

of the Prime Number Theorem and the Linnik Theorem:

- how often the values of f(x) at integral points in X are almost prime?

- can one effectively solve the congruence equation f(x)=b (mod q)

with f(x) being almost prime?

We discuss a solution to these questions when X is a homogeneous

variety (e.g, a quadratic surface).

We prove that no Riemannian manifold quasiisometric to

complex hyperbolic plane can have a better curvature pinching. The proof

uses cup-products in $L^p$-cohomology.