Moduli and Invariants (19w5171)
The Casa Matemática Oaxaca (CMO) will host the "Moduli and Invariants" workshop in Oaxaca, from November 17, 2019 to November 22, 2019. This workshop brings together groups of researchers from different groups studying enumerative invariants from moduli spaces of curves. The workshop will discuss recent advances in Donaldson-Thomas theory, Gromov-Witten theory and equivariant extensions of such theories. The Casa Matemática Oaxaca (CMO) in Mexico, and the Banff International Research Station for Mathematical Innovation and Discovery (BIRS) in Banff, are collaborative Canada-US-Mexico ventures that provide an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. The research station in Banff is supported by Canada's Natural Science and Engineering Research Council (NSERC), the U.S. National Science Foundation (NSF), Alberta's Advanced Education and Technology, and Mexico's Consejo Nacional de Ciencia y Tecnología (CONACYT).
18
2019
48
17 Stunden 15 Minuten
18 Ergebnisse
55:53
1Cruz Morales, John AlexanderWe will show by "generic smoothness" that the big quantum cohomology ring of isotropic Grassmannians IG(2,2n) is generically semisimple but the small quantum cohomology ring is not. That non-semisimplicity leads to a decomposition of the small quantum cohomology ring that relates to a certain decomposition of the derived category of IG(2,2n) in a so-called Lefschetz exceptional collection. This is based on joint work with A. Mellit, A. Kuznetsov, N. Perrin and M. Smirnov.
2019Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
57:16
6Oh, JeongseokWe define a localised Euler class for isotropic sections, and isotropic cones, in SO(N) bundles. We use this to give an algebraic definition of Borisov-Joyce's sheaf counting invariants on Calabi-Yau 4-folds. When a torus acts, we prove a localisation result. This talk is based on the joint work with R. P. Thomas.
2019Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
1:00:33
4Bejleri, DoriThe moduli of stable log varieties or stable pairs (X,D) are the higher dimensional analogue of the compactified moduli of stable pointed curves. The existence of a proper moduli space has been established thanks to the last several decades of advancements in the minimal model program. However, the notion of a family of stable pairs remains quite subtle, and in particular a deformation-obstruction theory for these moduli is not known. When the boundary divisor D is empty, Abramovich and Hassett gave an approach to stable varieties that replaces X with an associated orbifold. They show in this setting that the quite subtle notion of family of stable varieties becomes simply a flat family of the associated orbifolds. We extend this approach to the case where there is a nonempty but reduced boundary divisor D with the hopes of producing a deformation-obstruction theory for these moduli spaces. As an application we show that this approach leads to functorial gluing morphisms on the moduli spaces, generalizing the clutching and gluing morphisms that describe the boundary strata of the moduli of curves. This is joint work with G. Inchiostro.
2019Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
59:59
4Janda, FelixI will present a moduli space of "logarithmic R-maps" (joint work with Q. Chen and Y. Ruan), which together with its virtual cycles are the key ingredient toward an approach to proving many "mirror" conjectures about higher genus Gromov-Witten invariants of quintic threefolds (joint work with S. Guo and Y. Ruan).
2019Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
1:01:01
3Webb, RachelA driving question in Gromov-Witten theory is to relate the invariants of a complete intersection to the invariants of the ambient variety. In genus-zero this can often be done with a ``twisted theory,'' but this fails in higher genus. Several years ago, Chang-Li presented the moduli space of p-fields as a piece of the solution to the higher-genus problem, constructing the virtual cycle on the space of maps to the quintic 3-fold as a cosection localized virtual cycle on a larger moduli space (the space of p-fields). Their result is analogous to the classical statement that the Euler class of a vector bundle is the class of the zero locus of a generic section. I will discuss work joint with Qile Chen and Felix Janda where we extend Chang-Li's result to a more general setting, a setting that includes standard Gromov-Witten theory of smooth orbifold targets and quasimap theory of GIT targets. This work is joint with Qile Chen and Felix Janda.
2019Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
1:04:00
6Zhang, MingFor a large class of GIT quotients X=W//G, Ciocan-Fontanine-Kim-Maulik and many others have developed the theory of epsilon-stable quasimaps. The conjectured wall-crossing formula of cohomological epsilon-stable quasimap invariants for all targets in all genera has been recently proved by Yang Zhou. In this talk, we will introduce permutation-equivariant K-theoretic epsilon-stable quasimap invariants with level structure and prove their wall-crossing formulae for all targets in all genera. In physics literature, these invariants are related to the 3dN=2 supersymmetric gauge theories studied by Jockers-Mayr, and the wall-crossing formulae can be interpreted as relations between invariants in the UV and the IR phases of the 3d gauge theory. It is based on joint work in progress with Yang Zhou.
2019Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
1:00:31
4Kool, Martijn2019Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
59:10
Shen, YefengGromov-Witten invariants of Calabi-Yau one-folds (including elliptic curves and elliptic orbifold curves) are quasimodular forms. This can be proved using tautological relations and some ordinary differential equations in the theory of quasimodular forms, with minimal calculations. Such a method is also applicable to the Fan-Jarvis-Ruan-Witten theory of simple elliptic singularities. This allow us to prove the LG/CY correspondence for all CY one-folds using Cayley transformation of quasimodular forms, where GW/FJRW invariants are coefficients of Fourier/Taylor expansions of the same quasimodular forms. This talk is based on joint work with Jie Zhou, and Jun Li, Jie Zhou.
2019Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
52:32
4Bryan, JimThe Hilbert scheme parameterizing n points on a K3 surface X is a holomorphic symplectic manifold with rich properties. In the 90s it was discovered that the generating function for the Euler characteristics of the Hilbert schemes is related to both modular forms and the enumerative geometry of rational curves on X. We show how this beautiful story generalizes to K3 surfaces with a symplectic action of a group G. Namely, the Euler characteristics of the "G-fixed Hilbert schemes” parametrizing G-invariant collections of points on X are related to modular forms of level |G| and the enumerative geometry of rational curves on the stack quotient [X/G] . These ideas lead to some beautiful new product formulas for theta functions associated to root lattices. The picture also generalizes to refinements of the Euler characteristic such as χy genus and elliptic genus leading to connections with Jacobi forms and Siegel modular forms.
2019Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
51:49
1Chen, QileLogarithmic Gromov-Witten theory virtually counts the number of holomorphic curves with prescribed tangency condition along boundary divisors. In this talk I will introduce a variant of logarithmic maps called the punctured logarithmic maps. They naturally appear in a generalization of the gluing formulas of Li-Ruan and Jun Li. The punctured invariants play the role of relative invariants in these classical gluing formulas. They extend logarithmic Gromov-Witten theory by allowing negative tangency conditions with boundary divisors. This talk is based on a joint work with Dan Abramovich, Mark Gross and Bernd Siebert.
2019Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
58:50
1Kiem, Young-HoonFor the moduli of derived category objects or the partial desingularizations of the moduli stack of semistable sheaves on Calabi-Yau 3-folds, there are no perfect obstruction theories but only semi-perfect obstruction theories. While a semi-perfect obstruction theory is sufficient for the construction of virtual cycles in Chow groups, it seems insufficient for virtual structure sheaves. In this talk, I will introduce the notion of an almost perfect obstruction theory, which lies in between a semi-perfect obstruction theory and an honest perfect obstruction theory. I will show that an almost perfect obstruction theory enables us to construct the virtual structure sheaf and hence K-theoretic virtual invariants. Examples of DM stacks with almost perfect obstruction theories include the Inaba-Lieblich moduli spaces of simple gluable perfect complexes and the partial desingularizations of moduli stacks of semistable sheaves on Calabi-Yau 3-folds. We thus obtain K-theoretic Donaldson-Thomas invariants of derived category objects and K-theoretic generalized Donaldson-Thomas invariants. Based on a joint work with Michail Savvas.
2019Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
47:58
1Shoemaker, MarkIn this talk we reframe a collection of well-known comparison results in genus-zero Gromov-Witten theory in order to relate these to integral transforms between derived categories. This implies that various comparisons between Gromov-Witten and FJRW theory are compatible with the integral structure introduced by Iritani. We conclude with a proof that a version of the LG/CY correspondence relating quantum D-modules with Orlov's equivalence is implied by a version of the crepant transformation conjecture.
2019Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
1:01:11
1Donagi, RonThe Geometric Langlands Conjecture (GLC) for a curve C and a group G is a non-abelian generalization of the relation between a curve and its Jacobian. It claims the existence of Hecke eigensheaves on the moduli of G-bundles on C. The parabolic GLC is a further extension to curves with punctures. After explaining and illustrating the conjectures, I will outline an approach to proving them using non-abelian Hodge theory. A key geometric ingredient is the locus of wobbly bundles: bundles that are stable but not very stable. If time allows, I will discuss two instances where this program has been implemented recently: GLC for G=GL(2) and genus 2 curves (with T. Pantev and C. Simson), and parabolic GLC for P1 with marked points (with T. Pantev).
2019Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
52:51
Donovan, WillThe derived symmetries associated to a 3-fold admitting an Atiyah flop may be organised into an action of the fundamental group of a sphere with three punctures, thought of as a stringy Kaehler moduli space. I extend this to general flops of irreducible curves on 3-folds in joint work with M. Wemyss. This uses certain deformation algebras associated to the curve and its multiples, with applications to Gopakumar-Vafa invariants.
2019Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
56:03
1Borisov, DennisThe moduli stacks of sheaves on Calabi-Yau four-folds carry -2-shifted symplectic structures (Pantev, Toen, Vezzosi, Vaquie). Viewing these stacks as objects in differential geometry, one can construct Lagrangian foliations relative to these symplectic structures, such that quotients by the foliations are perfectly obstructed derived stacks, equipped with globally defined -1-shifted potentials, whose critical loci are the original moduli stacks. This is a joint work with A.Sheshmani and S-T.Yau.
2019Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
58:07
2Dumitrescu, OliviaIn 2014 Gaiotto conjectured a Lagrangian correspondence between holomorphic Lagrangian of opers in the Dolbeault moduli space of Higgs bundles and the de Rham moduli space of holomorphic connections. The conjecture was solved in 2016 for holomorphic opers in paper with Fredrickson, Kydonakis, Mazzeo, Mulase and Neitzke. By a similar analysis method, Collier and Wentworth, extended the correspondence for more general Lagrangians consisting of stable points. In my talk, I will present an algebraic geometry description of the Lagrangian correspondence of Gaiotto, based on the work of Simpson.
2019Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
58:39
3Gholampour, AminGiven a virtually smooth quasi-projective scheme M, and a morphism from M to a nonsingular quasi-projective variety B, we show it is possible to find an affine bundle M′/M that admits a perfect obstruction theory relative to B. We study the resulting virtual cycles on the fibers of M′/B and relate them to the image of the virtual cycle [M]vir under refined Gysin homomorphisms. Our main application is when M is a moduli space of stable codimension 1 sheaves on a nonsingular projective surface or Fano threefold.
2019Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
58:37
6Behrend, KaiWe study non-commutative projective varieties in the sense of Artin-Zhang, which are given by non-commutative homogeneous coordinate rings, which are finite over their centre. We construct moduli spaces of stable modules for these, and construct a symmetric obstruction theory in the CY3-case. This gives deformation invariants of Donaldson-Thomas type. The simplest example is the Fermat quintic in quantum projective space, where the coordinates commute only up to carefully chosen 5th roots of unity. We explore the moduli theory of finite length modules, which mixes features both of the Hilbert scheme of commutative 3-folds, and the representation theory of quivers with potential. This is mostly a report on the work of Yu-Hsiang Liu, with contributions by myself and Atsushi Kanazawa.
2019Banff International Research Station (BIRS) for Mathematical Innovation and Discovery