L3

Topology can be generalised in at least two directions: pointless

topology, leading ultimately to topos theory, or noncommutative

geometry. The former has the advantage that it also carries a logical

structure; the latter captures quantum settings, of which the logic is

not well understood generally. We discuss a construction making a

generalised space in the latter sense into a generalised space in the

former sense, i.e. making a noncommutative C*-algebra into a locale.

This construction is interesting from a logical point of view,

and leads to an adjunction for noncommutative C*-algebras that extends

Gelfand duality.

The talk will begin with a brief account of the construction of string topology operations. I will point out some mysteries with the formulation of these operations, such as the role of (moduli) of surfaces, and pose some questions. The remainder of the talk will address these issues. In particular, I will sketch some ideas for a higher-dimensional version of string topology. For instance, (1) I will describe an E_{d+1} algebra structure on the (shifted) homology of the free mapping space H_*(Map(S^d,M^n)) and (2) I will outline how to obtain operations H_*(Map(P,M)) -> H_*(Map(Q,M)) indexes by a moduli space of zero-surgery data on a smooth d-manifold P with resulting surgered manifold Q.

The second talk will present conjectural motivic generalizations

of ADHM sheaf invariants as well as their wallcrossing formulas.

It will be shown that these conjectures yield recursive formulas

for Poincare and Hodge polynomials of moduli spaces of Hitchin

pairs. It will be checked in many concrete examples that this recursion relation is in agreement with previous results of Hitchin, Gothen, Hausel and Rodriguez-Villegas.

The first talk will present a construction of equivariant

virtual counting invariants for certain quiver sheaves on a curve, called ADHM sheaves. It will be shown that these invariants are related to the stable pair theory of Pandharipande and Thomas in a specific stability chamber. Wallcrossing formulas will be derived using the theory of generalized Donaldson-Thomas invariants of Joyce and Song.

We report on joint work with Arend Bayer on the space of stability conditions for the canonical bundle on the projective plane.

We will describe a connected component of this space, generalizing and completing a previous construction of Bridgeland.

In particular, we will see how this space is related to classical results of Drezet-Le Potier on stable vector bundles on the projective plane. Using this, we can determine the group of autoequivalences of the derived category. As a consequence, we can identify a $\Gamma_1(3)$-action on the space of stability conditions, which will give a global picture of mirror symmetry for this example.

In the second hour we will give some details on the proof of the main theorem.

A categorification of cycle class maps consists to define

realization functors from constructible motivic sheaves to other

categories of coefficients (e.g. constructible $l$-adic sheaves), which are compatible with the six operations. Given a field $k$, we

will describe a systematic construction, which associates,

to any cohomology theory $E$, represented in $DM(k)$, a

triangulated category of constructible $E$-modules $D(X,E)$, for $X$

of finite type over $k$, endowed with a realization functor from

the triangulated category of constructible motivic sheaves over $X$.

In the case $E$ is either algebraic de Rham cohomology (with $char(k)=0$), or $E$ is $l$-adic cohomology, one recovers in this way the triangulated categories of $D$-modules or of $l$-adic sheaves. In the case $E$ is rigid cohomology (with $char(k)=p>0$), this construction provides a nice system of $p$-adic coefficients which is closed under the six operations.