SR1

The Siegel-Walfisz theorem gives an asymptotic estimate for the number of primes in an arithmetic progression, provided the modulus of the progression is small in comparison with the length of the progression. Counting primes is harder when the modulus is not so small compared to the length, but estimates such as Linnik's constant and the Brun-Titchmarsh theorem give us some information. We aim to look in particular at upper bounds for the number of primes in such a progression, and improving the Brun-Titchmarsh bound.

A K3 surface of degree two can be seen as a double cover of the complex projective plane, ramified over a nonsingular sextic curve. In this talk we explore two different methods for constructing explicit projective models of threefolds admitting a fibration by such surfaces, and discuss their relative merits.

The theory of C*-algebras provides a good realisation of noncommutative topology. There is a dictionary relating commutative C*-algebras with locally compact spaces, which can be used to import topological concepts into the C*-world. This philosophy fails in the case of homotopy, where a more sophisticated definition has to be given, leading to the notion of asymptotic morphisms.

As a by-product one obtains a generalisation of Borsuk's shape theory and a universal boundary map for cohomology theories of C*-algebras.

On a manifold there is the graded algebra of polyvector fields with its Lie-Schouten bracket, and the module of de Rham differentials with exteriour differentiation. This package is called a "calculus". The moduli space of sheaves (or derived category objects) on a Calabi-Yau threefold has a kind of "virtual calculus" on it, at least conjecturally. In particular, this moduli space has virtual de Rham cohomology groups, which categorify Donaldson-Thomas invariants, at least conjecturally. We describe some attempts at constructing such a virtual calculus. This is work in progress.

Both parts will deal with spherical objects in the bounded derived

category of coherent sheaves on K3 surfaces. In the first talk I will

focus on cycle theoretic aspects. For this we think of the Grothendieck

group of the derived category as the Chow group of the K3 surface (which

over the complex numbers is infinite-dimensional due to a result of

Mumford). The Bloch-Beilinson conjecture predicts that over number

fields the Chow group is small and I will show that this is equivalent to

the derived category being generated by spherical objects (which

I do not know how to prove). In the second talk I will turn to stability

conditions and show that a stability condition is determined by its

behavior with respect to the discrete collections of spherical objects.

This talk will follow the paper 0911.4708.

We shall explain what is the weighted fundamental lemma and how it is related to the truncated Hitchin fibration.

We shall explain what is the weighted fundamental lemma and how it is related to the truncated Hitchin fibration.

We shall explain what is the weighted fundamental lemma and how it is related to the truncated Hitchin fibration.

We shall explain what is the weighted fundamental lemma and how it is related to the truncated Hitchin fibration.