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GBMS Theme 6: Invariants from Moduli Spaces

Date: 24 - 27 Aug 2015

Venue: Western Gateway Building, UCC

Schedule

Theme 6 talks will take place in room G04 of Western Gateway Building. All plenary talks will be in room 107. Talks by Theme 4 invited speakers will take place in room G02.

Monday 24th August:

08:50-09:00 Opening

09:00 - 10:00

room G04

Alessio Corti  - Mirror Symmetry and Fano Orbifolds

Abstract: I discuss results and conjectures on mirror symmetry for Fano orbifolds and applications to the classification of Fano orbifolds, with special emphasis on the case of surfaces.

10:00 - 10:30 Coffee & Tea

10:30 - 11:30

room G04

Ionuț Ciocan-Fontanine -Wall-crossing in quasimap theory

 

Abstract: Quasimap theory is concerned with curve counting on certain GIT quotients. In fact, one has a family of curve counting theories, depending on the linearization in the GIT problem. It may be viewed as one possible mathematical incarnation of the GLSM in the geometric phases - the linearization is the FI parameter of the GLSM.

I will present a wall-crossing formula at the level of virtual classes (in all genera) as the size of the linearization changes, as well as some of its numerical consequences. This is joint work with Bumsig Kim.

 

11:30 - 12:30

room G04

Andrew Morrison - Motivic classes of generalized Kummer schemes via relative power structures

Abstract: We describe the cohomology of moduli spaces of points on schemes over Abelian varieties and give explicit calculations for schemes in dimensions less that three. The construction of Gulbrandsen allows one to consider virtual motives in dimension three. In particular we see a new proof of his conjectures on the Euler numbers of generalized Kummer schemes recently proven by Shen. Joint work with Junliang Shen.

12.30 - 14.00   Lunch

14.00 - 15.00

room G04

Sven Meinhardt -

About the nature of Donaldson-Thomas invariants for quivers with potential

Abstract: The aim of the talk is to show how Donaldson-Thomas sheaves are related to intersection complexes of quiver moduli spaces and vanishing cycle functors. In particular, Donaldson-Thomas theory can be used to provide an easy (recursive) algorithm to compute the intersection cohomology of quiver moduli spaces. This result has been obtained in a joint work with Ben Davison and Markus Reineke.

15.00 - 16.00

room G04

Gregory Sankaran -

 Homotopy invariants of toroidal compactifications

 Abstract: I will describe recent and less recent work of several people (mostly other people) on the behaviour of basic topological invariants, such as cohomology and the fundamental group, for moduli spaces compactified by toroidal methods.

 

16.00 - 16.30  Coffee & Tea

16.30 - 17.30

room 107

 
Martin J. Lindsay Plenary talk 

KMS-Symmetry and quantum Markov semigroups 

Tuesday 25th August:

09.00 - 10.00

room G04

Sergey Mozgovoy - Refined curve counting on some local 3CY varieties

I will discuss refined Donaldson-Thomas and Pandharipande-Thomas invariants of the 3CY varieties associated to some surfaces. The surfaces under consideration are minimal resolutions of Kleinian singularities. I will show how the problem can be reduced to counting representations of some quivers with superpotentials or counting representations of preprojective algebras. Then I will show how the latter problem is solved.

10.00 - 10.30  Coffee & Tea

10.30 - 11.30

room G04

 Barbara Fantechi  - On the rigidity of moduli stacks of pointed curves and of their coarse moduli spaces.

Abstract: Paul Hacking proved that the moduli stacks of complex pointed stable curves are rigid. Jointly with Alex Massarenti we try and extend this result to fields of arbitrary characteristic, and apply the results to studying automorphisms groups. We also study deformations of their coarse moduli spaces.

11.30 - 12.30

room G04

 Yuan-Pin Lee - Gromov--Witten theory and variation of Hodge structure under conifold transitions for Calabi-Yau threefolds.

 Abstract: The moduli of Calabi--Yau threefolds are generally believed to be connected by a geometric process called transition, which is roughly speaking degeneration followed by small resolution. In this talk, I will explain a phenomenon of partial exchange of A model (Gromov--Witten theory) and B model (variation of Hodge structure) when a Calabi--Yau threefold undergoes a conifold transition. This suggests a possibility of an "A+B theory" which is invariant under transitions. 

 

This talk is based on joint work with H.-W. Lin and C.-L. Wang.

12.30 - 14.00  Lunch

14.00 - 15.00

room G04

 Felix Janda -A formula for the double ramification cycle

Abstract: In the cohomology of the moduli space of smooth curves it is natural to consider the class of the locus of curves admitting a map to the projective line with specified ramification at zero and infinity. The double ramification cycle is an extension of this class to the Deligne-Mumford compactification of stable curves. Recently, generalizing work of Hain, Pixton has given a conjectural formula for the double ramification cycle in terms of tautological generators. We will present a proof of this formula.

15.00 - 16.00

room G04

 Nicola Pagani - Wall-crossing on the universal compactified Jacobian: the theta divisor

Abstract: The moduli space of line bundles on a curve can be non-compact even when the worst singularities of the curve are nodes. A natural compactification is obtained by adding stable rank-1 torsion-free sheaves: such compactification depends on the choice of a polarization on the nodal curve. Similarly, when compactifying the universal Jacobian over the moduli space of stable curves, one obtains a family of compact birational moduli spaces that depend on a polarization parameter. In this talk I will present a wall-crossing formula that describes how the theta divisor varies with this parameter. This is a joint work with Jesse Kass (South Carolina).
16.00 - 16.30  Coffee & Tea

16.30 - 17.30

room 107

  Dana Scott  - Plenary talk 

 Setoids/Modest Sets/PERs: Adding and using types with a type-free λ-Calculus

19.30  Dinner at the Cornstore, Cornmarket Street

Wednesday 26th August:

09.00 - 10.00

room 107

 Maciej Dunajski Plenary talk 

 Quartics, Sextics and Beyond

10.00 - 10.30  Coffee & Tea

10.30 - 11.30

room G04

 Aaron Bertram -Birational Geometry of Moduli Spaces of Sheaves on the Projective Plane via Stability Conditions

 

Abstract: When the minimal model program is applied to a moduli space, it is natural to ask whether the resulting birational models are themselves moduli spaces. In the case of moduli spaces of stable coherent sheaves on some algebraic surfaces, the answer is an emphatic yes, with moduli of Bridgeland-stable objects in the derived category arising as the birational models. In this talk I want to survey some of the known results, particularly focusing on the case of the projective plane.

 

11.30 - 12.30

room G04

 Andrew Kresch -

Conic bundles that are not stably rational

A specialization technique introduced recently by C. Voisin has sparked progress on questions of stable rationality. Varieties now known not to be stably rational include very general quartic double solids (Voisin) and very general quartic threefolds (Colliot-Thélène and Pirutka). This talk, representing joint work with B. Hassett and Y. Tschinkel, shows how the technique may be applied to conic bundles over rational surfaces.

12.30 - 14.00  Lunch

14.00 - 15.00

room G04

 Jan Manschot - Sheaves on surfaces and generalized Appell functions

Abstract: I will discuss generating functions of topological invariants of moduli spaces of semi-stable sheaves over a complex surface. For rational and ruled surfaces, these generating functions can be explicitly determined for arbitrary rank, Chern classes and polarization. To classify and study the building blocks of these generating functions, the notion of Appell functions with signature (m,n) is introduced. For n=1, these functions reduce to the known class of Appell functions with multiple variables or higher level.

15.00 - 16.00

room G02

  Ian McIntosh (Theme 4) - Equivariant minimal surfaces in the complex hyperbolic plane and their Higgs bundles.

Abstract: One approach to studying the space of irreducible representations of a surface group (fundamental group of a closed orientable surface) into the isometry group of a real or complex hyperbolic space is to parameterise these by Higgs bundles. It is well-known that the Higgs bundle equations are equivalent to the equations for an equivariant harmonic map into the hyperbolic space. But there are "too many" Higgs bundles: there is one parameterisation for every choice of conformal structure on the surface. One way to try to rectify this is to look for the best conformal structures by focussing on minimal surfaces. I will describe progress in this direction for maps into the complex hyperbolic plane, and how minimal surfaces can tell us more about the properties of a representation.

16.00 - 16.30   Coffee & Tea

16.30 - 17.30

room 107

Mark Gross  - Plenary talk 

 Logarithmic Gromov-Witten Invariants

Thursday 27th August:

09.00 - 10.00

room G04

 Alexander Schmitt - Semistable quiver sheaves

Abstract: We will survey the theory of quiver sheaves with special emphasis on the notion of semistability coming from Geometric Invariant Theory. In particular, we will present our recent boundedness results.
10.00 - 10.30  Coffee & Tea

10.30 - 11.00

room 107

IMS opening

11.00 - 12.00

room G04

 Martijn Kool -

Moduli spaces of rank 2 sheaves on toric 3-folds

The double dual of a torsion free sheaf on a 3-fold X is a reflexive sheaf R. When the rank is 2 and X is toric, we find generating functions of Euler characteristics of Quot schemes of R. As an application, we calculate generating functions of Euler characteristics of moduli spaces of rank 2 stable sheaves with fixed c_1, "low" c_2, and arbitrary c_3. When X is Calabi-Yau, the same generating functions arise from Donaldson-Thomas type invariants defined by virtual localization. Joint work with A. Gholampour and B. Young.

12.00 - 13.00

room 107

 Barbara Fantechi (IMS talk) -

Counting curves on algebraic varieties

Abstract: Enumerative geometry is one of the oldest parts of mathematics, with entry-level problems that can be explained to the layman (given four lines in space, how many lines meet/intersect all of them?). Yet in recent decades stunning progress has come thanks to a synergy of algebraic and complex geometry, string theory and function theory. In this talk we give a very partial overview, highlighting the interplay between concrete problems and theoretical advances.

13.00 End of Theme 6.  IMS conference and Theme 4 continue
 

Theme 6 Dinner

The dinner for Theme 6 is on Tuesday 25 August, 19:30 at the Cornstore Restaurant 40A Cornmarket Street, Cork.

For all Speakers of Theme 6: The dinner is free of charge and you are already booked by default. Please let the Theme organizers know if you cannot come to the dinner so that we can adjust the booking numbers.

For all delegates of Theme 6, who are not speakers: you are very welcome to join the dinner on a self-paying basis. Please let the Theme organizers know if you would like to come to the dinner so that we can adjust the booking numbers.

If you enjoy walking to the dinner (about 30 minutes walk), please meet up at 18:50 in the WGB atrium. Weather permitting we will leave WGB at 19:00 sharp and walk to the Cornstore Restaurant. Alternatively you can reach the restaurant by bus number 208 or by taxi.

 

Organizers

  • Anca Mustata (UCC)
  • Andrei Mustata (UCC)

Speakers Include (*=TBC)

  • Aaron Bertram (University of Utah)
  • Ionut Ciocan-Fontanine (University of Minnesota)
  • Alessio Corti (Imperial College London)
  • Barbara Fantechi (SISSA Trieste)
  • Mark Gross (Cambridge University)
  • Felix Janda (ETH Zürich)
  • Andrew Kresch (University of Zürich)
  • Martijn Kool (Utrecht University)
  • Yuan-Pin Lee (University of Utah)
  • Jan Manschot (Trinity College Dublin)
  • Sven Meinhardt (University of Wuppertal)
  • Andrew Morrison (ETH Zürich)
  • Sergey Mozgovoy (Trinity College Dublin)
  • Nicola Pagani (University of Liverpool)
  • Gregory Sankaran (University of Bath)
  • Alexander Schmitt (Freie Universität Berlin)

Sponsors

This is event is supported by Foundation Compositio Mathematica.

Learn more about the George Boole 200 programme at: www.georgeboole.com