Dinner Sunday at 19:30
11:00-12:15 Mellit I
15:45-17:00 Oprea I
17:30-18:45 Farkas I
9:30-10:30 Mellit II
16:15-17:15 Oprea II
17:45-18:45 Farkas II
11:00-12:15 Mellit III
15:45-17:00 Janda I
17:30-18:45 Farkas III
9:30-10:45 Janda II
11:00-12:15 Oprea III
Titles and Abstracts
G. Farkas (Humboldt Univ.)
Talk I: Topological invariants of groups via Koszul modules
Talk II: Green's Conjecture: a new approach via Koszul modules
I will discuss and prove an algebraic statement concerning the vanishing of the Koszul modules associated to any subspace K inside the second exterior product of a complex vector space. This statement, which turns out to be equivalent to Mark Green's Conjecture on syzygies of canonical curves (initially proven by Claire Voisin), has many interesting topological applications of which I will discuss (i) a universal upper bound on the nilpotence index of the fundamental group of any compact Kaehler manifold and (ii) a bound on the length of the nilpotence index on the Torelli groups associated to the moduli space of curves and (iii) an explicit description of the Cayely-Chow form of the Grassmannian G(2,n). This is joint work with M. Aprodu, S. Papadima, C. Raicu and J. Weyman. The first lecture will be devoted to the more topological applications, the second to the new approach to Green's Conjecture.
Talk III: Quadric rank loci on moduli spaces of curves and K3 surfaces
Given two vector bundles E and F on a variety X and a morphism from Sym^2(E) to F, we compute the cohomology class of the locus in X where the kernel of this morphism contains a quadric of prescribed rank. Our formulas have many applications to moduli theory: (i) we find a simple proof of Borcherds' result that the Hodge class on the moduli space of polarized K3 surfaces of fixed genus is of Noether-Lefschetz type, (ii) we construct an explicit canonical divisor on the Hurwitz space parametrizing degree k covers of the projective line from curves of genus 2k-1, (iii) we provide a closed formula for the Petri divisor on the moduli space of curves consisting of canonical curves which lie on a rank 3 quadric and (iv) construct myriads of effective divisors of small slope on M_g. Joint with R. Rimanyi.
F. Janda (Univ. of Michigan)
Talks I-II: On higher genus Gromov-Witten invariants of quintic threefolds
Gromov-Witten invariants intersection-theoretically count of curves
of fixed genus and degree inside a smooth manifold X. The case that X
is a quintic threefold has been of significant interest for both
mathematicians and string theorists.
In the first lecture, I survey conjectures from physics and the
progress made in mathematics regarding the computation of Gromov-Witten
invariants of quintic threefolds, including recent progress made in
work with S. Guo and Y. Ruan.
In the second lecture, I will discuss the foundations of the recent
work, and in particular a compactification of the moduli space of
stable maps with p-fields, as developed in joint work with Q. Chen and
S. Lysenko (Univ. of Lorraine)
Talk: On the Whittaker category in the metaplectic geometric Langlands theory
First, I will recall the Satake equivalence for a reductive group G from the geometric Langlands program. I will also recall the description of the Whittaker category Whit for G as a module over the geometric analog of Hecke algebra. Then I'll give a brief introduction to the geometric metaplectic Langlands theory. The rest of the talk is the discussion of the analog of the above two results in the metaplectic case. First, we associate to G and some "metaplectic data" the metaplectic Langlands dual group. Then I will present some partial results towards the proof of the Lurie-Gaitsgory conjecture. The latter attaches to the above data a big quantum group U_q(G^L) (Lustig's version with q-divided powers) and predicts that the corresponding Whittaker category Whit defined via geometry is canonically equivalent to the category of finite-dimensional representations of U_q(G^L). The category of representations of the corresponding small graded quantum group is known to be equivalent to certain category of factorizable sheaves FS (by a work of Bezrukavnikov, Funkelberg and Schechtman). We construct a functor from Whit to FS corresponding conjecturally to the restriction of representations from the big to the small quantum group. We will also describe the semi-simplification of Whit as a module over the metaplectic Hecke algebra.
A. Mellit (Univ. of Vienna)
Talks I-III: Counting over finite fields
The main goal of this mini-course is to explain the basic tools for counting various kinds of linear algebra objects over finite fields. We will begin by introducing Hall algebras, and on the example of one-loop quiver we will show how Hall-Littlewood polynomials arise in this context. Then we will talk about counting parabolic bundles. Finally, as an application, we will show how counting results are used to calculate the Betti numbers of moduli spaces.
D. Oprea (UC San Diego)
Talks I-III: Verlinde numbers, strange duality and tautological integrals over Hilbert schemes of points
Recently there has been progress in evaluating the generating series of Segre integrals of tautological vector bundles over the Hilbert schemes of points on surfaces. In rank 1, the Segre series were the subject of a conjecture of Lehn from 1999. Quite surprisingly, in arbitrary rank, the Segre series are also conjecturally related to the Verlinde-type generating series of Euler characteristics of tautological line bundles over the Hilbert scheme. This conjectural relationship can be interpreted in the context of the strange duality for moduli spaces of sheaves. It is motivated by efforts to extend the arguments that established strange duality for curves to the surface context. In these lectures I will explore this circle of ideas, and explain some of the conjectures and results.