The aim of this school/workshop is to give an introduction to conformal blocks, their construction and use as research tools and objects in different branches of algebraic geometry and topology, in particular moduli spaces of algebraic curves and of vector bundles on curves.



Four mini-courses of 5 hours each will be held by the following speakers:

Prakash Belkale (Univ. North Carolina at Chapel Hill)

Angela Gibney (Univ. of Georgia at Athens)

Gregor Masbaum (Univ. Paris VI at Jussieu)

Aaron Pixton (Princeton University)

We have also scheduled a 45' daily question time, in which people can ask any of the speakers for clarifications, examples/counterexamples, etc.






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.



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.



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.



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.





Host institution: Dipartimento di Matematica "G. Castelnuovo" - "Sapienza" Università di Roma



PRIN 2009 "Moduli spaces and Lie theory"





Foundation Compositio Mathematica








GRIFGA-Gruppo di ricerca Italo-Francese in geometria algebrica




IRMAR - Université de Rennes 1







FIRB 2010 "Low-dimensional geometry and topology"





 The school/workshop will be held at the Department of Mathematics "Guido Castelnuovo" of "Sapienza" Università di Roma, in Aula di Consiglio (first floor).




How to get there. 


It is possible for PhD-students and young post-docs to apply for funding for lodging guaranteed by our sponsors. Subscribe to the conference on the registration page (top left of this homepage) to apply.

The deadline for applying for funding has expired.

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