Next Meeting

Topics:

Hemisystems, partial quadrangles, association schemes, and flocks Tim Penttila Colorado State University As yet unpublished work by a number of authors has shown that cometric Q-antipodal association schemes can be constructed from hemisystems of generalized quadrangles, and that every generalized quadrangle arising from a flock in odd characteristic has a hemisystem. It was known over thirty years ago that hemisystems give partial quadrangles, and strongly regular graphs, but only one example of a hemisystem was known until this millennium. How things have changed! This talk surveys this area, with emphasis on the two results above.

Applications of finite geometry to extremal graph theory Jason Williford University of Wyoming Extremal graph theory officially began in 1940 with the proof of Turan's theorem. This theorem states that the greatest number of edges in an n-vertex graph with Kt as a subgraph is attained by the complete (t-1)-partite graph that is as regular as possible. Erdos, Stone and Simonovits later proved that this answer is asymptotically correct when Kt is replaced by a graph H of chromatic number t, provided that t>2. By contrast, little is known about the case when the forbidden subgraph H is bipartite. In many cases where the answer is known, it is because of constructions using finite fields and geometries. We will discuss these cases and related open problems which may benefit from finite geometry.