L3

In the talk I plan to overview several constructions for finite dimensional represenations of DAHA: construction via quantization of Hilbert scheme of points in the plane (after Gordon, Stafford), construction via quantum Hamiltonian reduction (after Gan, Ginzburg), monodromic construction (after Calaque, Enriquez, Etingof). I will discuss the relations of the constructions to the conjectures from the first lecture.

In this talk we show how the computation of the group of components of Prym varieties of spectral covers leads to cohomological results on the moduli space of stable bundles originally due to Harder-Narasimhan. This is joint work with Christian Pauly.

In this talk I will describe Toda's results on deformations of the category Coh(X) of coherent sheaves on a complex manifold X. They come from deformations of X as a complex manifold, non-commutative deformations, and gerby deformations (which can all be interpreted as deformations of X as a generalized complex manifold). Toda also described how to deform Fourier-Mukai equivalences, and I will present some examples coming from mirror SYZ fibrations.

In topos-valid point-free topology there is a good analogue of regular measures and associated measure theoretic concepts including integration. It is expressed in terms of valuations, essentially measures restricted to the opens. A valuation $m$ is $0$ on the empty set and Scott continuous, as well as satisfying the modular law $$ m(U \cup V) + m(U \cap V) = m(U) + m(V). $$

Of course, that begs the question of why one would want to work with topos-valid point-free topology, but I'll give some general justification regarding fibrewise topology of bundles and a more specific example from recent topos work on quantum foundations.

The focus of the talk is the valuation locale, an analogue of hyperspaces: if $X$ is a point-free space (locale) then its valuation locale $VX$ is a point-free space whose points are the valuations on $X$. It was developed by Heckmann, by Coquand and Spitters, and by myself out of the probabilistic powerdomain of Jones and Plotkin.

I shall discuss the following results, proved in a draft paper "A monad of valuation locales" available at http://www.cs.bham.ac.uk/~sjv/Riesz.pdf:

• V is a strong monad, analogous to the Giry monad of measure theory.
• There is a Riesz theorem that valuations are equivalent to linear functionals on real-valued maps.
• The monad is commutative: this is a categorical way of saying that product valuations exist and there is a Fubini theorem.

The technical core is an analysis of simple maps to the reals. They can be used to approximate more general maps, and provide a means to reducing the calculations to finitary algebra. In particular the free commutative monoid $M(L)$ over a distributive lattice $L$, subject to certain relations including ones deriving from the modular law, can be got as a tensor product in a semilattice sense of $L$ with the natural numbers. It also satisfies the Principle of Inclusion and Exclusion (in a form presented without subtraction).