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Workshop Programme

for period 11 - 15 April 2011

Derived categories

11 - 15 April 2011

Timetable

Monday 11 April
10:00-11:00 Registration
11:00-11:25 Morning Coffee
11:25-11:30 Welcome from Ben Mestel (INI Deputy Director)
11:30-12:30 Cautis, S (Columbia, NY)
  Categorified Heisenberg actions on Hilbert schemes Sem 1
 

I will describe an action of a quantized Heisenberg algebra on the (derived) categories of coherent sheaves on Hilbert schemes of ALE spaces (crepant resolutions of C^2/G). This action essentially lifts the actions of Nakajima and Grojnowski on the cohomology of these spaces. (joint with Tony Licata)

 
12:30-13:30 Lunch at Wolfson Court
15:00-16:00 Nakajima, H (Kyoto)
  Quiver varieties and tensor products Sem 1
 

I will explain how tensor products of representations of Yangian are realized in terms of geometry of quiver varieties.

 
16:00-16:30 Afternoon Tea
16:30-17:30 Hanany, A (Imperial College London)
  Trivertices and a corresponding class of hyperKahler spaces Sem 1
 

Given a graph with lines and 3-valent vertices, one can construct, using a simple dictionary, a Lagrangian that has N=2 supersymmetry in 3+1 dimensions. The vacuum moduli space of such a theory is well known to give moment map equations for a HyperKahler manifold.

We will discuss the class of hyperkahler manifolds which arise due to such a construction and present their special properties. The Hilbert Series of these spaces can be computed and turns out to be a function of the number of external legs and loops in the graph but not on its detailed structure. The corresponding SCFT consequence of this property indicates a crucial universality of many Lagrangians, all of which have the same dynamics.

The talk is based on http://arXiv.org/pdf/1012.2119

 
17:30-18:30 Welcome wine reception
18:45-19:30 Dinner at Wolfson Court (residents only)
Tuesday 12 April
10:00-11:00 Toda, Y (Tokyo)
  Stable pairs on local K3 surfaces. Sem 1
 

I give a formula which relates Euler characteristic of moduli spaces of stable pairs on local K3 surfaces to counting invariants of semistable sheaves on them. The formula generalizes Kawai-Yoshioka's formula for stable pairs with irreducible curve classes to arbitrary curve classes. I also propose a conjectual multi-covering formula of sheaf counting invariants which, combined with the main result, leads to an Euler characteristic version of Katz-Klemm-Vafa conjecture for stable pairs.

 
11:00-11:30 Morning Coffee
11:30-12:30 Pandharipande, R (Princeton)
  Why descendents? Sem 1
 

I will discuss several motivations and results concerning descendents in 3-fold sheaf theories.

 
12:30-13:30 Lunch at Wolfson Court
15:00-16:00 Mukai, S (Kyoto)
  K3 surfaces of genus 17 Sem 1
 

The moduli space M=M(2, h, 8) of semi-rigid vector bundles on a (polarized) K3 surface (S, h) of genus 17 is a K3 surface of genus 5. Moreover, the universal family gives an equivalence between the derived category of S and a twisted derived category of M. This equivalence induces us a rational map from S to the non-abelian Brill-Noether locus SU(2, K; 5F) of type II (see alg-geom/9704015) in the moduli space of 2-bundles on a curve of genus 5. We show that this map is an isomorphism when the modulus of (S, h) is general, using Thaddeus' formula. As a corollary the moduli space F17 of (S, h)’s is unirational.

 
16:00-16:30 Afternoon Tea
16:30-17:30 Bondal, A (Steklov Institute in Moscow & IPMU (Japan))
  Minuscule varieties and their degenerations Sem 1
 

We will give a description of toric degenerations of minuscule varieties in terms of moduli spaces of representations of quivers. We apply this to description of the structure of derived categories of coherent sheaves on minuscule varieties and to constructing Landau-Ginzburg models for them.

 
18:00-18:45 Wine reception at CUP Bookshop
18:45-19:45 Dinner at Wolfson Court (residents only)
Wednesday 13 April
10:00-11:00 Horja, P (Oklahoma State)
  Categorical Matrix Factorizations Sem 1
 

I will present a notion of matrix factorizations associated to an additive category endowed with a weak action of the integers. The role of the potential is played by a certain natural transformation compatible with the categorical weak action. Various applications of this framework to the understanding of Landau-Ginzburg models will be explored.

 
11:00-11:30 Morning Coffee
11:30-12:30 Iritani, H (Kyoto)
  Landau-Ginzburg/Calabi-Yau correspondence, analytic continuation and global mirror symmetry Sem 1
 

Under Landau-Ginzburg/Calabi-Yau correspondence, the quantum cohomology of a weighted projective hypersurface is analytically continued to the FJRW quantum ring of the associated Landau-Ginzburg model. Via the Gamma integral structure, we will see that the analytic continuation is induced from Orlov's equivalence between the derived category of coherent sheaves and the category of matrix factorizations. We also mention to a relation to mirror symmetry. This is based on joint work with Alessandro Chiodo and Yongbin Ruan.

 
12:30-13:30 Lunch at Wolfson Court
15:00-16:00 Kuznetsov, A (RAS)
  Exceptional collections on Grassmannians of classical groups Sem 1
 

I will describe a new approach to construction of exceptional collections on homogeneous varieties of semisimple algebraic groups. Using this approach I will construct exceptional collections of the expected length on all Grassmannians of classical groups. This is a joint work with Sasha Polishchuk.

 
16:00-16:30 Afternoon Tea
16:30-17:30 Zaslow, E (Northwestern)
  Ribbon Graphs and Mirror Symmetry Sem 1
 

Beginning with a ribbon graph with some extra structure, I will define a dg category, the "constructible plumbing model," which serves as a stand-in for the Fukaya category of the Riemann surface associated to the ribbon graph. When the graph has a combinatorial version of a torus fibration with section, I will define a one-dimensional algebraic curve, and prove that the dg category of vector bundles on the curve is equivalent to the constructible plumbing model, a version of homological mirror symmetry in one-dimension. I will also discuss the higher-dimensional case.

This talk is based on joint work with Nicolo' Sibilla and David Treumann.

 
19:30-22:00 Conference Dinner at Gonville & CaiusCollege
Thursday 14 April
10:00-11:00 Dimofte, T (Princeton)
  S-duality and mirror symmetry in Chern-Simons theory Sem 1
 

I will present joint work with S. Gukov on S-duality in three-dimensional Chern-Simons theories. Upon analytic continuation, the Chern-Simons "level" k can be taken to be an arbitrary complex number. Then, both Chern-Simons wavefunctions and certain operators that annihilate them acquire a certain symmetry under the inversion of k (S-duality). In terms of the operators, this can be understood in terms of mirror symmetry acting on branes in a hyperkahler space (a Hitchin moduli space).

 
11:00-11:30 Morning Coffee
11:30-12:30 Diaconescu, DE (Rutgers)
  BPS states, Donaldson-Thomas invariants and Hitchin pairs Sem 1
 

A string theoretic construction will be presented relating BPS states, Donaldson-Thomas invariants and the cohomology of the Hitchin system. In particular, the work of Hausel and Rodrgiguez-Villegas will be given a natural physical explanation based on geometric engineering.

This is work in collaboration with Wu-yen Chuang and Guang Pan.

 
12:30-13:30 Lunch at Wolfson Court
15:00-16:00 Shende, V (Princeton)
  Hilbert schemes of singular plane curves and HOMFLY homology of their links Sem 1
 

Intersecting a plane curve with the boundary of a small ball around one of its singularities yields a link in the 3-sphere. To any link may be attached a triply graded vector space, the HOMFLY homology. Taking its Euler characteristic with respect to a certain grading gives the HOMFLY polynomial, which in turn specializes variously to the Alexander polynomial, the Jones polynomial, and the other SU(n) knot polynomials.

We will present a conjecture recovering this invariant from moduli spaces attached to the singular curve. Specifically, we form the Hilbert schemes of points of the curve, and certain incidence varieties inside products of Hilbert schemes. Up to certain shifts of grading, we conjecture that the HOMFLY homology of the link of the singularity is the direct sum of the homologies of these spaces.

This talk presents joint work with J. Rasmussen and A. Oblomkov.

 
16:00-16:30 Afternoon Tea
16:30-17:30 Joyce, D (Oxford)
  D-manifolds, a new theory of derived differential geometry Sem 1
 

I describe a new class of geometric objects I call "d-manifolds". D-manifolds are a kind of "derived" smooth manifold, where "derived" is in the sense of the derived algebraic geometry of Jacob Lurie, Bertrand Toen, etc. The definition draws on ideas of Jacob Lurie, David Spivak. The original aim of the project, which I believe I have achieved, is to find the "right" definition of the Kuranishi spaces of Fukaya, Oh, Ohta and Ono, which is the geometric structure on moduli spaces of J-holomorphic curves in a symplectic manifold.

The definition of d-manifolds involves doing algebraic geometry over smooth functions (C-infinity rings); roughly speaking, a d-manifold is a differential-geometric analogue of a scheme with a perfect obstruction theory. D-manifolds form a strict 2-category dMan. It is a 2-subcategory of the larger 2-category of "d-spaces" dSpa. The definition does not involve localization of categories, so we have very good control of what 1-morphisms and 2-morphisms are.

The 2-categories dMan and dSpa have some very nice properties. All fibre products exist in dSpa, and a fibre product of d-manifolds is a d-manifold under weak transversality condition. For example, any fibre product of two d-manifolds over a manifold is a d-manifold. You can glue d-manifolds by equivalences of open d-submanifolds (a kind of pushout in dMan) provided the glued topological space is Hausdorff. There is a notion of "virtual cotangent bundle" of a d-manifold, which lives in a 2-category of virtual vector bundles, and a 1-morphism of d-manifolds is etale (a local equivalence) iff it induces an equivalence of virtual cotangent bundles. And so on.

There are also good notions of d-manifolds with boundary and d-manifolds with corners, and orbifold versions of all this, d-orbifolds.

D-manifolds and d-orbifolds have applications to moduli spaces and enumerative invariants in both differential and algebraic geometry. Almost any moduli space which is used to define some kind of counting invariant should have a d-manifold or d-orbifold structure. Any moduli space of solutions of a smooth nonlinear elliptic p.d.e. on a compact manifold has a d-manifold structure. Any C-scheme with a perfect obstruction theory has a d-manifold structure. In symplectic geometry, Kuranishi spaces and polyfold structures on moduli spaces of J-holomorphic curves induce d-orbifold structures. So much of Gromov-Witten theory, Donaldson-Thomas theory, Lagrangian Floer cohomology, Symplectic Field Theory,... can be rewritten in this language.

 
18:45-19:30 Dinner at Wolfson Court (residents only)
Friday 15 April
09:30-10:30 Soibelman, I (Kansas State)
  Integrable systems, Mirror Symmetry and Donaldson-Thomas invariants Sem 1
 

This talk is about applications of some ideas from Mirror Symmetry to complex integrable systems. It is a joint work with Maxim Kontsevich, and it is related to our theory of Donaldson-Thomas invariants for 3d Calabi-Yau categories. In particular, wall-crossing formulas lead to a construction of ``exponential Hodge structure of infinite type".

 
10:30-11:00 Morning Coffee
11:00-12:00 Bryan, J (British Columbia)
  On the motivic class of the commuting variety and related problems Sem 1
 

In 1960, Feit and Fine found a beautiful formula for the number of commuting n by n matrices over the finite field F_q. Their result can be reinterpreted as a formula for the motivic class of the commuting variety in the Grothendieck group. We will describe a simple new proof of their formula which allows us to generalize the result to several other settings with applications to motivic Donaldson-Thomas theory.

 
12:00-13:00 Lunch at Wolfson Court
13:00-14:00 Nagao, K (Kyoto)
  Wall-crossing of the motivic Donaldson-Thomas invariants Sem 1
 

We study the motivic Donaldson-Thomas invariants introduced by Kontsevich-Soibelman and Behrend-Bryan-Szendroi. A wall-crossing formula is proved for a certain class of mutations of quivers with potentials.

 
14:00-14:30 Afternoon Tea
14:30-15:30 Szendroi, B (Oxford)
  Motivic Donaldson-Thomas theory of the conifold Sem 1
 

The talk will explain the computation of the motivic refinement of Donaldson-Thomas theory and related enumerative theories on the resolved conifold geometry, in all chambers of the space of quiver stability conditions. The results are in full agreement with the expected answer from the refined topological vertex of Iqbal-Kozcaz-Vafa, and the wall-crossing computations of Jafferis-Chuang. This is joint work with Andrew Morrison and Kentaro Nagao.

 
18:45-19:30 Dinner at Wolfson Court (residents only)

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