Open University

The celebrated Szemeredi-Trotter theorem states that the maximum number of incidences between $n$ points and $n$ lines in the plane is $\mathcal{O}(n^{4/3})$, which is asymptotically tight.

Solymosi conjectured that this bound drops to $o(n^{4/3})$ if we exclude subconfigurations isomorphic to any fixed point-line configuration. We describe a construction disproving this conjecture. On the other hand, we prove new upper bounds on the number of incidences for configurations that avoid certain subconfigurations. Joint work with Martin Balko.

We will be joined by Dr Brigitte Stenhouse from the Open University to discuss harassment, particularly in academic settings. 

I shall report on recent progress, in Australia and Germany, on the empirical evaluation of special values of L-functions by minors of period matrices whose elements include Feynman integrals from diagrams with up to 20 loops. Previously such relations were known only for diagrams with up to 6 loops.
 

The entire history of mathematics in one hour, as illustrated by around 300 postage stamps featuring mathematics and mathematicians from across the world.

From Euclid to Euler, from Pythagoras to Poincaré, and from Fibonacci to the Fields Medals, all are featured in attractive, charming and sometimes bizarre stamps. No knowledge of mathematics or philately required.

I describe recent work with Pyber, Short and Szabo in which we study the `width' of a finite simple group. Given a group G and a subset A of G, the `width of G with respect to A' - w(G,A) - is the smallest number k such that G can be written as the product of k conjugates of A. If G is finite and simple, and A is a set of size at least 2, then w(G,A) is well-defined; what is more Liebeck, Nikolov and Shalev have conjectured that in this situation there exists an absolute constant c such that w(G,A)\leq c log|G|/log|A|. 
I will present a partial proof of this conjecture as well as describing some interesting, and unexpected, connections between this work and classical additive combinatorics. In particular the notion of a normal K-approximate group will be introduced.