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The Sarkisov Program

Abstract

When pursuing a classification program as broad as the Mori program - the classification of projective varieties up to birational equivalence - it is critical to have a series of guiding strategic schemes by which to approach the classification. For instance, (1) isolate some distinguished representatives to be considered the ‘best’ in each equivalence class, and (2) understand the relationships between the many possible best representatives. Towards scheme (2), in this talk I will give a visual introduction to the Sarkisov program - an elegant algorithm which factorises birational maps between best representatives (when the best representative is not unique) into a series of ‘elementary’ transformations (of which there are exactly 4 types).  

Purple Ramsey

Abstract

Determining the Ramsey numbers R(s,t) - or at least asymptotics bounds - is one of the foremost problems in combinatorics. This talk gives a quick introduction to the Ramsey numbers, along with novel variant of the problem, where an edge can be coloured both red and blue. We present a few results from upcoming work on this ‘purple’ Ramsey problem.