Oxford

In this talk we will introduce the (bounded) derived category of coherent sheaves on a smooth projective variety X, and explain how the geometry of X endows this category with a very rigid structure. In particular we will give an overview of a theorem of Orlov which states that any sufficiently ‘nice’ functor between such categories must be Fourier-Mukai.

In particular, some nice things about branch groups, whose subgroup structure  "sees" all actions on rooted trees.

In particular, some nice things about branch groups, whose subgroup structure "sees" all actions on rooted trees.

Motivated by an open conjecture in anabelian geometry, we investigate which arithmetic properties of the rationals are encoded in the absolute Galois group G_Q. We give a model-theoretic framework for studying absolute Galois groups and discuss in what respect orderings and valuations of the field are known to their first-order theory. Some questions regarding local-global principles and the transfer to elementary extensions of Q are raised.

Everything measurable about a quantum system, as modelled by a noncommutative operator algebra, is captured by its commutative subalgebras. We briefly survey this programme, and zoom in one specific incarnation: any bilinear associative function on the set of n-by-n matrices over a field of characteristic not two, that makes the same vectors orthogonal as ordinary matrix multiplication and gives the same trace as ordinary matrix multiplication, must in fact be ordinary matrix multiplication (or its opposite). Model-theoretic questions about the hypotheses and scope of this theorem are raised.

In this talk we will review some of the main ideas around Hamilton's program for the Ricci flow and see how they fit together to provide a proof of the Poincaré conjecture in dimension 3. We will then analyse this tools in the context of 4-manifolds.

I shall discuss joint work with Mladen Bestvina in which we prove that the group of simplicial automorphisms of the complex of free factors for a
free group $F$ is exactly $Aut(F)$, provided that $F$ has rank at least $3$. I shall begin by sketching the fruitful analogy between automorphism groups of free groups, mapping class groups, and arithmetic lattices, particularly $SL_n({\mathbb{Z}})$. In this analogy, the free factor complex, introduced by Hatcher and Vogtmann, appears as the natural analogue in the $Aut(F)$ setting of the spherical Tits building associated to $SL_n $ and of the curve complex associated to a mapping class group. If $n>2$, Tits' generalisation of the Fundamental Theorem of Projective Geometry (FTPG) assures us that the automorphism group of the building is $PGL_n({\mathbb{Q}})$. Ivanov proved an analogous theorem for the curve complex, and our theorem complements this. These theorems allow one to identify the abstract commensurators of $GL_n({\mathbb{Z}})$, mapping class groups, and $Out(F)$, as I shall explain.

In this talk, I develop the Hopf algebra of renormalization, as established by Connes and Kreimer. I then use the correspondence between commutative Hopf algebras and affine groups to show that the energy scale dependence of the renormalized theory can be expressed as a Maurer Cartan connection on the renormalization group.