L3

The curve complex on an orientable surface, introduced by William Harvey about 30 years ago, is the abstract simplicial complex whose vertices are isotopy classes of simple close curves. A set of vertices forms a simplex if they can be represented by pairwise disjoint elements. The mapping class group of S acts on this complex in a natural way, inducing a homomorphism from the mapping class group to the group of automorphisms of the curve complex. A remarkable theorem of Nikolai V. Ivanov says that this natural homomorphism is an isomorphism. From this fact, some algebraic properties of the mapping class group has been proved. In the last twenty years, this result has been extended in various directions. In the joint work with Ferihe Atalan, we have proved the corresponding theorem for non-orientable surfaces: the natural map from the mapping class group of a nonorientable surface to the automorphism group of the curve compex is an isomorphism. I will discuss the proof of this theorem and possible applications to the structure of the mapping class groups.

If $X$ is a Fano variety with canonical bundle $O(-k)$, its derived category

has a semi-orthogonal decomposition (I will say what that means)

\[ D(X) = \langle O(-k+1), ..., O(-1), O, A \rangle, \]

where the subcategory $A$ is the "interesting piece" of $D(X)$. In the previous talk we saw that $A$ can have very rich geometry. In this talk we will see a less well-understood example of this: when $X$ is a smooth cubic in $P^5$, $A$ looks like the derived category of a K3 surface. We will discuss Kuznetsov's conjecture that $X$ is rational if and only if $A$ is geometric, relate it to Hassett's earlier work on the Hodge theory of $X$, and mention an autoequivalence of $D(Hilb^2(K3))$ that I came across while studying the problem.

   I will present a decidability result for theories of large fields of algebraic numbers, for example certain subfields of the field of totally real algebraic numbers. This result has as special cases classical theorems of Jarden-Kiehne, Fried-Haran-Völklein, and Ershov.

   The theories in question are axiomatized by Galois theoretic properties and geometric local-global principles, and I will point out the connections with the seminal work of Ax on the theory of finite fields.

We are interested in measure theory and integration theory that ¯ts into the
o-minimal context. Therefore we introduce the following de¯nition:
Given an o-minimal structure M on the ¯eld of reals and a measure ¹ de¯ned on the
Borel sets of some Rn, we call ¹ M-tame if there is an o-minimal expansion of M such
that for every parameter family of functions on Rn that is de¯nable in M the family of
integrals with respect to ¹ is de¯nable in this o-minimal expansion.
In the ¯rst part of the talk we give the de¯nitions and motivate them by existing and
many new examples. In the second one we discuss the Lebesgue measure in this context.
In the ¯nal part we obtain de¯nable versions of important theorems like the theorem of
Radon-Nikodym and the Riesz representation theorem. These results allow us to describe
tame measures explicitly.
1

I will introduce the notion of a nilsequence, which is a kind of

"higher" analogue of the exponentials used in classical Fourier analysis. I

will summarise the current state of our understanding of these objects. Then

I will discuss a variety of applications: to solving linear equations in

primes (joint with T. Tao), to a version of Waring's problem for so-called

generalised polynomials (joint with V. Neale and Trevor Wooley) and to

solving certain pairs of diagonal quadratic equations in eight variables

(joint work with L. Matthiesen). Some of the work to be described is a

little preliminary!