L3

I will describe how to build a noncommutative ring which dictates

the process of resolving certain two-dimensional quotient singularities.

Algebraically this corresponds to generalizing the preprojective algebra of

an extended Dynkin quiver to a larger class of geometrically useful

noncommutative rings. I will explain the representation theoretic properties

of these algebras, with motivation from the geometry.

In this talk I will look at a definition of the energy-momentum for the dynamical horizon of a black hole. The talk will begin by examining the role of a special class of observers at null infinity determined by Bramson's concept of frame alignment. It is shown how this is given in terms of asymptotically constant spinor fields and how this framework may be used together with the Nester-Witten two form to give a definition of the Bondi mass at null infinity.

After reviewing Ashtekar's concept of an isolated horizon we will look at the propagation of spinor fields and show how to introduce spinor fields for the horizon which play the role of the asymptotically constant spinor fields at null infinity, giving a concept of alignment of frames on the horizon. It turns out that the equations satisfied by these spinor fields give precisely the Dougan-Mason holomorphic condition on the cross sections of the horizon, together with a simple propagation equation along the generators. When combined with the Nester-Witten 2-form these equations give a quasi-local definition of the mass and momentum of the black hole, as well as a formula for the flux across the horizon. These ideas are then generalised to the case of a dynamical horizon and the results compared to those obtained by Ashtekar as well as to the known answers for a number of exact solutions.

I will associate, to every pair of smooth transversal

Lagrangian spheres in a symplectic manifold having vanishing first Chern

class, its Floer cohomology groups. Hamiltonian isotopic spheres give

rise to isomorphic groups. In order to define these Floer cohomology

groups, I will make a key use of symplectic field theory.

For the integers $a$ and $b$ let $P(a^b)$ be all partitions of the

set $N= {1,..., ab}$ into parts of size $a.$ Further, let

$\mathbb{C}P (a^b)$ be the corresponding permutation module for the

symmetric group acting on $N.$ A conjecture of Foulkes says

that $\mathbb{C}P (a^b)$ is isomorphic to a submodule of $\mathbb{C}P

(b^a)$ for all $a$ not larger than $b.$ The conjecture goes back to

the 1950's but has remained open. Nevertheless, for some values of

$b$ there has been progress. I will discuss some proofs and further

conjectures. There is a close correspondence between the

representations of the symmetric groups and those of the general

linear groups, via Schur-Weyl duality. Foulkes' conjecture therefore

has implications for $GL$-representations. There are interesting

connections to classical invariant theory which I hope to mention.

The representation theory of symmetric groups starts with

the permutation modules. It turns out that the annihilator of a

permutation module can be described explicitly in terms of the

combinatorics of Murphy's cellular basis of the group algebra of the

symmetric group in question. In fact, we will show that the

annihilator is always a cell ideal. This is recent joint work with K.

Nyman.

I will push Schanuel's conjecture in four directions: defining a dimension

theory (pregeometry), blurred exponential functions, exponential maps of

more general groups, and converses. The goal is to explain how Zilber's

conjecture on complex exponentiation is true at least in a "geometric"

sense, and how this can be proved without solving the difficult number

theoretic conjectures. If time permits, I will explain some connections

with diophantine geometry.

We survey the classification of structures interpretable in o-minimal theories in terms of thorn-minimal types. We show that a necessary and sufficient condition for such a structure to interpret a real closed field is that it has a non-locally modular unstable type. We also show that assuming Zilber's Trichotomy for strongly minimal sets interpretable in o-minimal theories, such a structure interprets a pure algebraically closed field iff it has a global stable non-locally modular type. Finally, if time allows, we will discuss reasons to believe in Zilber's Trichotomy in the present context