Prakash BELKALE
Title: Gauss-Manin representation of conformal block local systems.
Abstract: Conformal blocks give projective local systems on moduli spaces of curves with marked points. One can ask if they are "realizable in geometry", i.e., as local subsystems of suitable Gauss-Manin local systems of cohomology of families of smooth projective varieties.
We will discuss (in genus 0) the proof of Gawedzki et al's conjecture that Schechtman-Varchenko forms are square integrable (this was proved first for sl(2) by Ramadas). Together with the flatness results of Schechtman-Varchenko, and the work of Ramadas, one obtains the desired realization and a unitary metric on conformal blocks.
Angela GIBNEY
Title: Conformal blocks divisors and the birational geometry of the moduli space of curves
Abstract: In these lectures I will present the combinatorial tools one needs to use first Chern classes of vector bundles of conformal blocks, after Fakhruddin, to study the cone of nef divisors of the moduli spaces of stable pointed curves. I will discuss what we have learned about conformal blocks divisors, as well as some open problems about them in relation to the Mori Dream Space Conjecture and Level-Rank duality.
Gregor MASBAUM
Title: Integral TQFT and applications to the monodromy of conformal blocks
Abstract: The space of conformal blocks on a smooth complex curve carries a projective representation of the mapping class group of the underlying topological surface. This representation is part of a Topological Quantum Field Theory (TQFT) in the sense of Atiyah and Segal.
In the first part of these lectures, I plan to sketch a construction of this TQFT using skein theory, which allows one in particular to write down explicit matrices for any given mapping class expressed as a word in Dehn twists. (The relevant skein theory will be developed from scratch and no previous knowledge of skein theory will be assumed.)
In the second part, I will then discuss an integral refinement of this TQFT constructed in joint work with Gilmer again using skein theory. This construction shows in particular that in favorable situations the (skein-theoretic version of the) space of conformal blocks contains a natural mapping class group invariant lattice of full rank defined over a ring of algebraic integers. If time remains, I will give some applications of this integrality property of the representation to questions about the mapping class group.
Aaron PIXTON
Title: Intersection theory on the moduli space of curves
Abstract: The Chow ring of an algebraic variety encodes information about how its subvarieties intersect each other. In the case of the moduli space of curves, the full structure of the Chow ring is not well understood. I will primarily talk about a subring of the Chow ring known as the tautological ring; this is the subring generated by those classes that arise most naturally in geometry.
After reviewing some of the basic features of the moduli space of curves and constructions of algebraic cycles on it, I will discuss the current state of knowledge about the structure of the tautological ring. This will include a quick survey of the many geometric approaches that have been used to produce relations in the tautological ring.
