# Abstracts, slides and videos

Here is the list of talks with their abstracts, slides (if the colour of the title is green) and whiteboard slide (if the colour of the title is blue). Very soon there will be the possibility to watch the videos in the Youtube channel of VBAC2013

A. Boralevi. Matrices of constant rank and instanton bundles.

Given a complex vector space $V$ of dimension $n$, one can look at $d$-dimensional linear subspaces $A$ in $\wedge^2(V)$, whose elements have constant rank $r$. The natural interpretation of $A$ as a vector bundle map yields some restrictions on the values that $r$, $n$ and $d$ can attain. I will show an effective bound on $d$ in the case of co-rank 2, and then concentrate on the 4-dimensional case, for which I will give a method that allows one to prove the existence of new examples. The technique involves instanton bundles of charge 2 and 4 and the derived category of $\mathbb{P}^3$, and provides also an explanation for what once used to be the only known example, by Westwick. These are joint works with D. Faenzi, E. Mezzetti, J. Buczynski and G. Kapustka.

We define actions of certain Heisenberg algebras on the Hilbert schemes of points on ALE spaces. This lifts constructions of Nakajima and Grojnowski from cohomology to K-theory and (derived) categories of coherent sheaves. This action can be used to define Lie algebra actions (using categorical vertex operators) and subsequently braid group actions and knot invariants.

M. Cirafici. Defects in cohomological gauge theory and Donaldson-Thomas invariants.

Certain aspects of Donaldson-Thomas theory on a Calabi-Yau can be described in terms of generalized instantons of a certain six-dimensional gauge theory. In this setup we introduce a class of defects which generalize surface defects in four dimensional gauge theories. These defects are associated with divisors. We discuss how the generalized instanton moduli space is modified and the possibility of a generalization of the Donaldson-Thomas invariants. In the case of the affine space the proposals can be conveniently rephrased in terms of the representation theory of a certain quiver.

P. Dalakov. Donagi-Markman cubic for the ramified Hitchin system.

The infinitesimal period map for an algebraic completely integrable hamiltonian system is encoded in a section of the third symmetric power of the cotangent bundle to the base of the system, the Donagi-Markman cubic. We present some work in progress on the explicit description of the cubic for the ramified Hitchin system.

D.E. Diaconescu. Parabolic refined invariants and Macdonald polynomials.

This is work in progress with Wu-yen Chuang, Ron Donagi and Tony Pantev building a string theoretic framework for the conjecture of Hausel, Letellier and Rodriguez-Villegas on the cohomology of character varieties with marked points. Their formula is identified with a Gopakumar-Vafa expansion in the refined stable pair theory of local orbifold curves, which is related by geometric engineering to K-theoretic invariants of nested Hilbert schemes. In particular MacDonald polynomials appear naturally in this framework via Haiman's geometric construction based on the isospectral Hilbert scheme. Supporting evidence is obtained by localization computations of parabolic refined invariants on the conifold via the equivariant index defined by Nekrasov and Okounkov.

G. Faonte. A comparison between dg categories and infinity categories.

The aim of the talk is a comparison between the notion of pretriangulated dg categories in the sense of Kapranov-Bondal [2] and stable infinity categories in the sense of Lurie [4]. I will sketch how to associate to a dg category two infinity categories, that we should call the big and small nerve, and how to prove [4] the equivalence of those construction. I will then prove that if a dg category is pretriangulated then the associated nerve is a stable infinity category. The last part of the talk will be dedicated to a construction of an A-infinity nerve for A-infinity categories that generalize the small nerve construction. References: [1] G. Faonte, The simplicial nerve of an A-infinity category". [2] A.I. Bondal, M. Kapranov, Enhanced Triangulated Categories". [3] J. Lurie, Higher Topos Theory". [4] J. Lurie, Higher Algebra".

M. Finkelberg. Laumon spaces and Macdonald polynomials.

We give a new geometric interpretation of the Macdonald polynomials in terms of cohomology of the Laumon moduli spaces with coefficients in the sheaves of differential forms twisted by line bundles. This is a joint work with A.Braverman and J.Shiraishi.

E. Franco. Higgs bundles over elliptic curves.

In this talk we will describe $G$-Higgs bundles over an elliptic curve $X$. A key result on our study is that a Higgs bundle for $G = GL(n,C)$ is (semi)stable if and only if the underlying vector bundle is (semi)stable. This fact can be extended to arbitrary complex reductive Lie groups and allows us to use the Atiyah's description of semistable vector bundles over $X$. We will give an explicit description of the moduli spaces of $G$-Higgs bundles and the Hitchin map. This is joint work with O. Garcia-Prada and P. Newstead.

I will discuss algebraic deformation quantization of the Hilbert scheme of points in the plane and other Nakajima quiver varieties, and related Beilinson-Bernstein localization which provides an equivalent between certain categories of sheaves on the deformation quantization and representations of related algebras.

Using relative Hilbert schemes of points of linear systems on curves on surfaces refined curve counting invariants are introduced and studied. For toric surfaces they are related to Welschinger invariants and to refined curve counting invariants in tropical geometry. For a large class of toric surfaces it is shown that these can computed in terms of the action of a Heisenberg algebra on a Fock space.

We describe three stratifications on a parameter space of quiver representations: the first is a stratification coming from geometric invariant theory, the second is a Morse theoretic stratification coming from the symplectic structure on the parameter space and the third is the Shatz stratification by Harder--Narasimhan types (in all three cases, they depend on a choice of stability parameter and a choice of norm on a certain gauge group). Finally, we show these three stratifications agree.

M. Kool. Curves on surfaces.

The Hilbert scheme of curves in class $\beta$ on a smooth projective surface $S$ carries a natural virtual cycle. In many cases this cycle is zero (e.g. when $S$ has a non-zero holomorphic 2-form and $\beta$ is not sub-canonical). However, in these cases one can often remove part of the obstruction bundle and obtain a non-trivial reduced'' virtual cycle. Both cycles have interesting applications. (1) Both are related to Pandharipande-Thomas' stable pair invariants on the total space of the canonical bundle over $S$. (2) The reduced virtual cycle is related to Severi degrees and classical curve counting on $S$. (3) The non-reduced virtual cycle is related to the Seiberg-Witten invariants of $S$ (by work of Duerr-Kabanov-Okonek and Chang-Kiem).

M. Jardim. Commuting matrices and the Hilbert scheme of points on affine varieties.

We give a linear algebraic description of the Hilbert scheme of points on affine varieties which naturally extends Nakajima's representation of the Hilbert scheme of points on the affine plane. We also introduce extended monads and perfect extended monads in order to generalize the monadic description of ideal sheaves of 0-dimensional subschemes of projective spaces. As an application of our ideas, we use results from the literature on commuting matrices to show the Hilbert scheme of $c$ points on $\mathbb{C}^3$ is irreducible for $c\le 10$ and reducible for $c\ge30$.

M. Lelli-Chiesa. On a conjecture of Donagi and Morrison.

The Donagi-Morrison conjecture concerns the Brill-Noether theory of curves $C$ lying on an arbitrary K3 surface $S$. It predicts that any linear series of type $g^r_d$ on $C$ with a negative Brill-Noether number is contained in the restriction to $C$ of an effective divisor $D$ on $S$, which satisfies a number of properties. The conjecture was soon verified by Donagi and Morrison for $r=1$ and I will explain how the study of the stability of Lazarsfeld-Mukai bundles of rank 3 proves successful in proving it for $r=2$. For arbitrary $r$ greater than two, by means of coherent systems, I will underline how the conjecture is related with the existence of some special secant varieties to $C$.

The boson-fermion correspondence is a relationship between the bosonic and fermionic Fock spaces. It uses vertex operators to express bosons in terms of fermions and vice-versa. However, the precise relationship between the two depends on the choice of embedding of the Heisenberg algebra in the affine general linear Lie algebra. Such embeddings are parametrized by conjugacy classes in the symmectric group. Licata and Savage have given a geometric realization of the boson-fermion correspondence for the conjugacy class of the identity element (the so-called homogeneous realization). In this talk, we will discuss a similar construction for other classes, in particular the one corresponding to the principal realization. This construction is in terms of the equivariant cohomology of the Hilbert scheme of points in the plane and Nakajima quiver varieties.

A. Negut. Moduli of flags of sheaves and shuffle algebras.

A few years ago, Schiffmann-Vasserot and Feigin-Tsymbaliuk independently introduced an action of a certain algebra on the K-theory of the Hilbert scheme. This algebra is known by many names, including the elliptic Hall algebra, $U_q(\widehat{\widehat{gl}}_1)$, doubly deformed $W_{1+\infty}$, the Ding-Iohara algebra and the double shuffle algebra. In this talk, we use the language of universal bundles to define an action of the double shuffle algebra on the K-theory of the moduli space of rank $r$ sheaves on $\mathbb{C}^2$, and we identify certain distinguished generators of this algebra with some new correspondences that parametrize certain flags of sheaves. These correspondences are well-defined in the derived category, and we hope that this will lead to a categorical action of the shuffle algebra (which is yet to be defined).

Let $C$ be a curve with locally planar singularities, let $C^{[n]}$ be the Hilbert scheme of $n$ points on $C$, and let $V$ be the direct sum of the homology groups $H_*(C^{[n]})$ for different $n$. We describe 4 creation/annihilation operators acting on $V$ and show that these satisfy the commutation relations for a Heisenberg algebra. (The operators are similar to those defined by Nakajima for Hilbert schemes of a smooth surface.) As a consequence of this action we recover a recent result of Maulik-Yun and Migliorini-Shende, which describes each $H_*(C^{[n]})$ explicitly in terms of the homology of the compactified Jacobian of $C$.

A. Savage. Formal Hecke algebras and algebraic oriented cohomology theories.

Many constructions in geometric representation theory involve defining important algebras via geometric operations, such as convolution, on the (co)homology of certain algebraic varieties. Examples of varieties appearing prominently in these formulations include Hilbert schemes, flag varieties and quiver varieties. Often the algebra constructed depends on the particular cohomology theory used. Motivated by these constructions, we define a "formal (affine) Hecke algebra" associated to any formal group law. Formal group laws are associated to algebraic oriented cohomology theories. When specialized to the formal group laws corresponding to K-theory and (co)homology, our definition recovers the usual affine and degenerate affine Hecke algebras. However, other formal group laws (such as those corresponding to elliptic and cobordism cohomology theories) give rise to apparently new algebras with interesting properties. This is joint work with Alex Hoffnung, Jose Malagon-Lopez, and Kirill Zainoulline.

A. Sheshmani. DT theory of torsion sheaves and modular forms.

We study the Donaldson-Thomas invariants of the 2-dimensional stable sheaves in a smooth projective threefold. The DT invariants are defined via integrating over the virtual fundamental class when it exists. When the threefold is a K3 surface fibration we express the DT invariants of sheaves supported on the fibers in terms of the the Euler characteristics of the Hilbert scheme of points on the K3 surface and the Noether-Lefschetz numbers of the fibration. Using this we prove the modularity of the DT invariants for threefolds given as K3 fibrations as well as local $\mathbb{P}^2$ which was predicted in string theory. We develop a DT-theoretic conifold transition formula through which we compute the generating series for the invariants of Hilbert scheme of points for singular surfaces. We also use our geometric techniques to compute the generating series for DT invariants of threefolds given as complete intersections such as the quintic threefold. Finally if time permits I explain furt her application of this study such as deep connections between our torsion DT invariants and higher dimensional Knot theory as well as proof of Crepant resolution conjecture on the B-model side. This is joint work with Amin Gholampour.

R. Szabo. Instantons and curve counting.

I will give an overview of the computation of certain curve counting invariants of toric Calabi-Yau 3-folds and of toric surfaces using instanton techniques from maximally supersymmetric gauge theories.

A. Tikhomirov. Moduli of mathematical instantons with even $c_2$.

The problem of the irreducibility of the moduli space $I_n$ of rank-2 mathematical instanton vector bundles with $c_2=n$ was recently solved affirmatively by the speaker for odd $n$. (This result is published in Izvestija RAN: Ser. Mat. 76:5 (2012), 143–224.) In this talk we give the proof of the irreducibility of the space $I_n$ for even $n$.

M. Tommasini. The Hodge-Deligne polynomials of some moduli spaces of coherent systems.

After proving the existence of (non-split, non-degenerate) universal families of extensions of coherent systems (in the spirit of Lange), we are able to describe wallcrossing for some moduli spaces of coherent systems. This allows us to compute the Hodge-Deligne polynomials of the moduli spaces of coherent systems of type $(1,d,n)$ for $n=2,3$ and of type $(2,d,2)$ for $d$ odd big enough.

A. Zamora. GIT characterization of the Harder-Narasimhan filtration for finite dimensional representations of quivers.

In a moduli problem where we use Geometric Invariant Theory to take the quotient to get a moduli space, an unstable object gives a GIT unstable point in certain parameter space. To a GIT unstable point, Kempf associates a “maximally destabilizing” 1-parameter subgroup, and this induces a filtration of the object. We show that this filtration coincides with the Harder-Narasimhan filtration for finite dimensional representations of a finite quiver on vector spaces, using the construction of a moduli space for these objects given by King. This work is a continuation of previous work joint with T Gomez and I. Sols, where we show a similar correspondence for moduli problems related to sheaves with additional structure, now in a different moduli problem. Reference: A. Zamora, "On the Harder-Narasimhan filtration for finite dimensional representations of quivers" arxiv: 1210.5475. To appear in Geometriae Dedicata.