Homological Mirror Symmetry

Matthew Ballard: Generation in Characteristic p

Abstract: I will talk about how to construct generators for derived categories when working in nonzero characteristic. Parts are joint work with Alex Duncan and Patrick McFaddin and with Pat Lank.

Rodrigo Barbosa: On Coassociative ALE Fibrations and Higgs Bundles

Abstract: Pantev and Wijnholt conjectured that G2-manifolds fibered by ALE spaces could be constructed from solutions of a gauge theoretic system on the base of the fibration. As in the case of ALE-fibered Calabi-Yau threefolds, the system is defined by (a Riemannian version of) Hitchin's equations. I will describe recent progress on this problem and how it ties up with Donaldson's program of building G2-manifolds fibered by coassociative K3 surfaces.

Enrique Becerra: On the Presentability of Artin Differentiable Stacks

Abstract: In this talk I will talk about the global quotient presentability of Artin stacks in the smooth setting.

Antoine Boivin: Compactification of Moduli Spaces of Quantum Toric Stacks

Abstract: A toric variety is a complex variety which is completely described by the combinatorial data of a fan of strongly convex rational (with respect to a lattice) cones. Due to this rationality condition, toric varieties are (equivariantly) rigid since if we deform a lattice, it can become dense. A solution to this problem is to consider a stacky generalization of toric varieties where the "lattice" is, in fact, a finitely generated subgroup of $\R^d$ (in the simplicial case as introduced by L. Katzarkov, E. Lupercio, L. Meersseman and A. Verjovsky). The goal of this talk is to explain the moduli spaces of quantum toric stacks and their compactification.

Laurent Côté: An Application of Symplectic Topology to Algebraic Geometry

Abstract: There is a notion of dimension for triangulated categories which was introduced by Rouquier. Orlov conjectured that the dimension of the bounded derived category of coherent sheaves of a smooth complex variety equals its ordinary Krull dimension. In joint work with Shaoyun Bai, we settled new cases of this conjecture using methods from symplectic geometry, in combination with homological mirror symmetry. I hope to explain this story and discuss prospects for further work in this direction.

Charles Doran: The Mirror Clemens-Schmid Sequence

Abstract: I will present a four-term exact sequence relating the cohomology of a fibration to the cohomology of an open set obtained by removing the preimage of a general linear section of the base. This exact sequence respects three filtrations, the Hodge, weight, and perverse Leray filtrations, so that it is an exact sequence of mixed Hodge structures on the graded pieces of the perverse Leray filtration. I claim that this sequence should be thought of as a mirror to the Clemens-Schmid sequence describing the structure of a degeneration and formulate a "mirror P=W" conjecture relating the filtrations on each side. Finally, I will present evidence for this conjecture coming from the K3 surface setting. This is joint work with Alan Thompson (arXiv:2109.04849).

David Favero: Homotopy Path Algebras

Abstract: I will consider a class of algebras, "Homotopy Path Algebras", naturally appearing in many contexts; e.g. algebraic topology, sheaf theory, and toric geometry (as full strong exceptional collections of line bundles). I will develop the general theory of such algebras and explain the connection to homological mirror symmetry, expanding on the ideas of Bondal and Fan-Lui-Truemann-Zaslow. This is based on joint work with Jesse Huang.

Kenji Fukaya: Gromov-Hausdorff Convergence of Filtered A Infinity Categories

Abstract: Floer homology over Novikov ring is not invariant under Hamiltonian isotopy. This dependence is important in its application to Hamiltonian dynamics and there are several interesting works on it recently. In this talk I explain how we include the multiplicative structure to it. I also explain how it is expected to be related to homological Mirror symmetry when we work over Novikov ring (formal deformation theory).

Sergei G. Gukov: Spectra, Dynamical Systems, and RG Flows

Abstract: What do Frobenius manifolds and dynamical systems have in common? To answer this question, this talk will rely on two components. The first component, based on an earlier work (arXiv: 1503.01474 and 1608.06638) will recast RG flows in the language of dynamical systems. One outcome of this reformulation is the conjecture that, under certain generic conditions that can be spelled out precisely, the spectrum of scaling dimensions in a CFT moves only in one direction, so that the values of \Delta_i increase. The second component of the talk (based on work in progress with L. Katzarkov, K.S. Lee, J. Svoboda, ...) aims to relate this statement to the semicontinuity of the spectrum due to Varchenko and Steenbrink. As a prerequisite, we will need a relation between the spectra of conformal dimensions and the Steenbrink spectra.

Andrew Harder: Mixed Hodge Structures Attached to Hybrid Landau–Ginzburg Models

Abstract: Given a Landau–Ginzburg model (Y,w) satisfying certain mild conditions there is a mixed Hodge structure, first constructed by Shamoto, and implicit in works of Katzarkov-Kontsevich-Pantev and Esnault-Sabbah-Yu. Essentially, this mixed Hodge structure is the limit mixed Hodge structure on the relative cohomology of Y with respect to the fibre of w over a point t, taken as t approaches infinity. If (Y,w) is mirror to a Fano variety X, the weight- and Hodge-graded pieces of this mixed Hodge structure carry information about the cohomology of X, similar to how the Hodge diamonds of mirror compact Calabi–Yau varieties are expected to reflect one another.

Shamoto's mixed Hodge structures have standard functorial properties related to open and closed subvarieties of Y, which can be seen either directly, or as a consequence of general operations on mixed Hodge modules. In this talk, I will discuss the functorial behaviour of Shamoto's mixed Hodge structure with respect to decompositions of the function w into sums, w = w_1 + w_2 satisfying mild conditions. Precisely, I will discuss a new spectral sequence relating the mixed Hodge structure attached to (Y,w) to the mixed Hodge structures attached to (Y,w_1) and (Y,w_2), and which is related to the perverse Leray filtration on the cohomology of Y coming from to the map (w_1,w_2). Time permitting, I will discuss several concrete applications of this spectral sequence.

R. Paul Horja: Discriminants and Toric Birational Geometry

Abstract: Given a Gorenstein toric singularity, I will explain a proposal for the B-side categories in toric homological mirror symmetry along the strata of the discriminant locus describing the singularities of the attached GKZ hypergeometric D-module. The conjectural construction of the web of associated spherical functors was proposed by Aspinwall, Plesser and Wang based on string theoretic arguments.

Gabriel Kerr: Lagrangian Intersections in Phase Tropical Hypersurfaces

Abstract: The phase tropical hypersurface is a topological manifold homeomorphic to a complex hypersurface in the complex torus. It admits a map to the tropical hypersurface which can be thought of as a combinatorial version of torus fibrations of T-duality. Building on this analogy as well as those from LG models, I will examine the combinatorial analogs of Lagrangian spheres, thimbles and intersections. The relation to the mirror matrix factorization categories will be discussed.

Maxim Kontsevich: Stability Structures in Holomorphic Morse-Novikov Theory

Abstract: I will talk about an elementary example illustrating the general program (by Y. Soibelman and myself) relating Floer theory for complex symplectic manifolds, quantization and resurgence. The specific question is about the space of morphisms between two specific branes in the cotangent bundle to a complex manifold. The first brane is the zero section endowed with a generic rank 1 local system, while the second brane is the graph of a closed holomorphic 1-form. In this case the whole story is reduced to Morse-Novikov theory and sheaf theory. Even in this relatively simple case one can see clearly delicate analytic questions dealing with convergence. All considerations generalize to the almost-complex case.

Kyoung-Seog Lee: Derived Categories and Motives of Moduli Spaces of Vector Bundles on Curves

Abstract: Derived categories and motives are important invariants of algebraic varieties invented by Grothendieck and his collaborators around the 1960s. In 2005, Orlov conjectured that they will be closely related and now there are several evidences supporting his conjecture. On the other hand, moduli spaces of vector bundles on curves provide attractive and important examples of algebraic varieties and there have been intensive works studying them. In this talk, I will discuss derived categories and motives of moduli spaces of vector bundles on curves and how they are related. Part of this talk is based on several joint works with I. Biswas, T. Gomez, H.-B. Moon and M. S. Narasimhan.

Chiu-Chu Melissa Liu: Homological Mirror Symmetry for Theta Divisors

Abstract: I will describe a version of global homological mirror symmetry for a theta divisor in a principally polarized abelian variety of arbitrary dimension, over the complex moduli of principally polarized and SYZ fibered abelian varieties. This is based on joint work in progress with Haniya Azam, Catherine Cannizzo, and Heather Lee.

Ernesto Lupercio: Progress on Quantum Toric Geometry

Abstract: Quantum toric geometry is to toric geometry what the quantum torus is to the usual torus. This talk is divided in two parts: (1) Introduction to Quantum Toric Geometry (where I introduce all the Dramatis Personae of the theory) and (2) Moduli and Mirrors (where I report on progress on the moduli properties and the mirror dualities). This is joint with Katzarkov, Meersseman and Verjovsky.

David Nadler: Functions on Commuting Stacks via Mirror Symmetry

Abstract: For a complex reductive group G, its commuting stack parametrizes pairs of commuting group elements up to conjugacy. One can also interpret the commuting stack as G-local systems on a torus. I'll explain joint work with Penghui Li and Zhiwei Yun that calculates global functions on the commuting stack via mirror symmetry, in particular Betti geometric Langlands.

Vivek Shende: Quantum Mirror Symmetry for the Conifold

Abstract: Consider the (all genus) open Gromov-Witten invariants for the conifold with a single Aganagic-Vafa brane. I will explain how to show, a priori (i.e. without first computing the invariants), that the generating series satisfies an operator equation, given by a skein-valued quantization of the mirror curve. Said equation gives a recursion which can be solved explicitly for the series. Time permitting, I will then derive the relationship between *colored* HOMFLYPT polynomials and Gromov-Witten invariants. [This talk presents joint work with Tobias Ekholm.]

Artan Sheshmani: Global Shifted Potentials for Moduli Stack of Sheaves over Calabi-Yau 4 Folds

Abstract: We report on series of joint works with Borisov, Katzarkov and Yau on construction of a globally defined shifted potential functionals over moduli stack of stable sheaves on Calabi-Yau 4 folds. Firstly we discuss globally defined shifted symplectic structures on DG Quot schemes of sheaves over DG manifolds. The theory of dg Quot schemes is developed so that it becomes a homotopy site, and the corresponding infinity category of stacks is equivalent to the infinity category of stacks, as constructed by Toen and Vezzosi, on the site of dg algebras whose cohomologies have finitely many generators in each degree. Stacks represented by dg schemes are shown to be derived schemes under this correspondence.

Then we discuss generalization of our construction over DG Quot stacks, or stacky quotient of of our DG schemes. We show that any derived scheme over C equipped with a (-2)-shifted symplectic structure, and having a Hausdorff space of classical points, admits a globally defined Lagrangian distribution as a dg C-infinity manifold. We will use this result to construct Lagrangian distributions on stable loci of derived Quot stacks. The main tool for proving our main theorem is a strictification result for Lagrangian distribution.

Finally we show that there are globally defined Lagrangian distributions on the stable loci of derived Quot-stacks of coherent sheaves on Calabi–Yau four-folds. Dividing by these distributions produces perfectly obstructed smooth stacks with globally defined (-1)-shifted potentials, whose derived critical loci give back the stable loci of smooth stacks of sheaves in global Darboux form. This report is based on joint papers: arXiv:1908.03021, arXiv:1908.00651, and arXiv:2007.13194.

Bernd Siebert: Canonical Wall Structures via Punctured Gromov-Witten Theory

Abstract: I will sketch the construction of consistent wall structures for (generalized) logarithmic Calabi-Yau pairs (X,D). This shows compatibility of the instrinsic mirror construction jointly with Mark Gross in arXiv:1909.07649, which is based on the direct definition of structure coefficients of the coordinate ring, and earlier constructions based on wall structures and rings of generalized theta functions.

The proof has been made possible by recent advances in punctured Gromov-Witten theory and gluing formulas, in joint work with Dan Abramovich, Qile Chen, and Mark Gross, as well as by my student Yixian Wu. This is joint work with Mark Gross (arXiv:2105.02502).

Yan Soibelman: Holomorphic Floer Theory, Quantum Wave Functions and Resurgence

Abstract: Finite dimensional exponential integrals over real cycles in the affine complex algebraic variety X can be interpreted in terms of exponential periods of X, or equivalently in terms of an isomorphism between twisted versions of the de Rham and Betti cohomology of X. This relation in turn can be upgraded in Holomorphic Floer Theory to an equivalence of the Fukaya category of the cotangent bundle of X and the category of holonomic modules over the deformation quantization of the cotangent bundle (a.k.a. holonomic DQ-modules).

Analytic behavior of the stationary phase expansion of the exponential integral with the phase f is controlled by the moduli space of pseudo-holomorphic discs with the boundaries on the pair of holomorphic Lagrangian submanifolds X and graph(df). In particular resurgence (a.k.a Borel resummability) of the expansion follows from exponential bounds on the virtual number of such discs.

In this talk I am going to recall the above story and to discuss its upgrade to the case of infinite-dimensional exponential integrals including e.g. the complexified Chern-Simons theory. An important mathematical structure emerging in this upgrade is the one of the quantum wave function. The talk is based on the joint project with Maxim Kontsevich.

Josef Svoboda: Ẑ-invariants and Universal Abelian Cover

Abstract: For a plumbed 3-manifold M which is a rational homology sphere, we can study its invariants in terms of its universal abelian cover M^ab equipped with the action of the first homology group of M. M^ab can be naturally thought of as a link of an isolated singularity, so analytic invariants such as monodromy, spectrum or Poincaré series come into play. We will present how to use this point of view to study (GPPV) Ẑ-invariants of M. Work in progress with S. Gukov, L. Katzarkov and K.S. Lee.

Constantin Teleman: The Curved Cartan Complex Revisited

Abstract: The CCC is an effective device for computing (B-model) gauged algebras and categories in 3 dimensions. In this lecture I will (partially) explain its relation to the alternative computation from the Toda integrable system and the construction of Coulomb branches of polarized representations from 2D mirror symmetry.

Yixian Wu: A Splitting Formula for Punctured Gromov-Witten Invariants

Abstract: Punctured Gromov-Witten theory of Abramovich-Chen-Gross-Siebert is an extension of logarithmic Gromov-Witten theory in which marked points have a negative order of tangency with boundary divisors. It plays an important role in the intrinsic mirror symmetry construction of Gross and Siebert. The underlying combinatorial structures of punctured maps are encoded in tropical geometry. In this talk, I will present a formula reconstructing the punctured invariants under the operation of splitting along edges. It is a generalization of the degeneration formula of Jun Li in the logarithmic Gromov-Witten invariants setting and provides a new technique to compute Gromov-Witten invariants.

Tony Yue Yu: Non-archimedean Quantum K-theory and Gromov-Witten Invariants

Abstract: Motivated by mirror symmetry and the enumeration of curves with boundaries, it is desirable to develop a theory of Gromov-Witten invariants in the setting of non-archimedean geometry. I will explain our recent works in this direction. Our approach differs from the classical one in algebraic geometry via perfect obstruction theory. Instead, we build on our previous works on the foundation of derived non-archimedean geometry, the representability theorem and Gromov compactness. We obtain numerical invariants by passing to K-theory or motivic cohomology. We prove a list of natural geometric relations between the stacks of stable maps, directly at the derived level, with respect to elementary operations on graphs, namely, products, cutting edges, forgetting tails and contracting edges. They imply the corresponding properties of numerical invariants. The derived approach produces highly intuitive statements and functorial proofs. Furthermore, its flexibility allows us to impose not only simple incidence conditions for marked points, but also incidence conditions with multiplicities. Joint work with M. Porta.