Hodge theory and derived categories of cubic fourfolds

Richard Thomas
Imperial College London
September 16, 2014
Cubic fourfolds behave in many ways like K3 surfaces. Certain cubics - conjecturally, the ones that are rational - have specific K3s associated to them geometrically. Hassett has studied cubics with K3s associated to them at the level of Hodge theory, and Kuznetsov has studied cubics with K3s associated to them at the level of derived categories.

I will explain all this via some pretty explicit examples, and then I will explain joint work with Addington showing that these 2 notions of having an associated K3 surface coincide generically.

Generic K3 categories and Hodge theory

Daniel Huybrechts
University of Bonn
September 16, 2014
In this talk I will focus on two examples of K3 categories: bounded derived categories of (twisted) coherent sheaves and K3 categories associated with smooth cubic fourfolds. The group of autoequivalences of the former has been intensively studied over the years (work by Mukai, Orlov, Bridgeland and others), whereas the investigation of the latter has only just began. As a motivation, I shall recall Mukai's classification of finite groups of automorphisms of K3 surfaces and its more recent derived version which involves the Leech lattice.

Colouring graphs with no odd holes

Paul Seymour
Princeton University
September 22, 2014
The chromatic number \(k(G)\) of a graph \(G\) is always at least the size of its largest clique (denoted by \(w(G)\)), and there are graphs with \(w(G)=2\) and \(k(G)\) arbitrarily large. On the other hand, the perfect graph theorem asserts that if neither \(G\) nor its complement has an odd hole, then \(k(G)=w(G)\). (An ``odd hole" is an induced cycle of odd length at least five.) What happens in between?

Uniform words are primitive

Doron Puder
Member, School of Mathematics
September 23, 2014
Let \(G\) be a finite group, and let \(a\), \(b\), \(c\),... be independent random elements of \(G\), chosen at uniform distribution.
What is the distribution of the element obtained by a fixed word in the letters \(a\), \(b\), \(c\),..., such as \(ab\), \(a^2\), or \(aba^{-2}b^{-1}\)? More concretely, do these new random elements have uniform distribution?

In general, a word \(w\) in the free group \(F_k\) is called uniform if it induces the uniform distribution on every finite group \(G\). So which words are uniform?

Gamma Ray Bursts from a Different Angle: The Sequel

David Eichler
Ben Gurion University
September 23, 2014

A classic problem posed by long gamma ray bursts (GRB) is that the energy output requires gravitational energy release so deep within the host star that the prompt gamma rays should, upon naive consideration, have been obscured. It is suggested that photons emitted along the direction of the emitting plasma's motion are indeed geometrically blocked by optically thick baryonic matter, and that we usually see the photons that are emitted nearly backward in the frame of the emitting plasma. Many puzzling observations concerning GRB then fall into place.