Homological Mirror Symmetry

Maxim Kontsevich, Institut des Hautes Etudes Scientifiques: Non-Holomorphic Deformations of Landau-Ginzburg Models I,II

With a non-singular complex variety Y together with a holomorphic function W one can associate a sheaf of differential graded Lie algebras on Y consisting of polyvector fields with the differential given by the commutator with W. The formal germ of the derived moduli space is smooth and finite-dimensional if the critical locus is compact, Y is a Kahler and carries a non-vanishing holomorphic volume element. A part of the derived moduli space can be interpreted as moduli of holomorphic deformations of the pair (Y,W), but the whole derived moduli space does not have a direct holomorphic interpretation. Such spaces are of great interests as they carry a weakened form of the Frobenius manifold structure, and are expected to describe genus 0 Gromov-Witten invariants for general symplectic manifolds. I'll talk about Fukaya-theoretic and Hodge-theoretic aspects of non-holomorphically deformed LG models, including a generalization of theory of spectra for isolated singularities (joint work in progress with D.Auroux and L.Katzarkov).  Video

Benjamin Gammage, Harvard University: Introduction to 3d Mirror Symmetry

3d mirror symmetry is a proposed duality between holomorphic symplectic manifolds with Hamiltonian group actions. In this talk we describe a new formulation of this theory in terms of an equivalence between a dual pair of 2-categories (of A- and B-type, respectively), categorifying the A- and B-brane categories of homological (2d) mirror symmetry. We describe this 2-categorical equivalence in the abelian case, where it entails a spectral decomposition of the 2-category of spherical functors. This is based on joint work with Justin Hilburn and Aaron Mazel-Gee.  Video

Yan Soibelman, Kansas State University: Holomorphic Floer Theory, Wall-Crossing Structures and Chern-Simons Theory

Name "Holomorphic Floer theory" (HFT) was suggested by Maxim Kontsevich and myself in 2014 for the area of mathematics devoted to questions related to Fukaya categories of complex symplectic manifolds. Many problems in HFT has their "A-side" (i.e. Fukaya category side) and "B-side" (which is the deformation quantization of the same complex symplectic manifold). One incarnation of the relation between A and B sides is our  generalized Riemann-Hilbert correspondence reported at one of the HMS workshops in Miami several years ago. A very special case of the RH-correspondence  leads to the Picard-Lefschetz wall-crossing formulas for exponential integrals. One of the aims  of my talk is to explain how  these wall-crossing formulas and  related wall-crossing structures arise from a pair of holomorphic Lagrangian subvarieties of a complex symplectic manifold and how they are related to  resurgence of perturbative series in  Chern-Simons theory. This "Chern-Simons wall-crossing structure" can be  described in different ways, e.g. in terms of the finite-dimensional geometry of certain K_2-Lagrangian subvarieties of a complex torus or in terms of a conjectural Hodge structure of infinite rank.  Video  Slides

Maxim Jeffs, Harvard University: Functorial Mirror Symmetry for very Affine Hypersurfaces

In this talk I will start by explaining some conjectures of Auroux, which state that homological mirror symmetry for very affine hypersurfaces should respect certain natural symplectic operations (as functors between wrapped Fukaya categories). These conjectures concern compatibility between mirror symmetry for a very affine hypersurface and its complement, itself also a very affine hypersurface. In joint work with Benjamin Gammage, we find that the complement of a very affine hypersurface has in fact two natural mirrors, one of which is a derived scheme. I will explain our proof of Auroux’s conjectures, after they have been modified to take this derived structure into account, using sectorial techniques for gluing Fukaya categories.  Video

Heather Lee, University of Washington: Homological Mirror Symmetry for Theta Divisors

In this talk we will describe the correspondence under homological mirror symmetry (HMS) between the complex moduli of theta divisors in principally polarized abelian varieties and the Kaehler moduli of the mirror Landau-Ginzburg models.  We aim at studying global HMS over the entire moduli space.  For abelian varieties in complex dimension 2, theta divisors are genus 2 curves, and our work generalizes Cannizzo's thesis, which proves a HMS result for a one parameter family of genus 2 curves in the moduli space. This is based on joint work with Haniya Azam, Catherine Cannizzo, and Chiu-Chu Melissa Liu.  Video  Slides

Maxim Kontsevich, Institut des Hautes Etudes Scientifiques: Non-Holomorphic Deformations of Landau-Ginzburg Models I,II

With a non-singular complex variety Y together with a holomorphic function W one can associate a sheaf of differential graded Lie algebras on Y consisting of polyvector fields with the differential given by the commutator with W. The formal germ of the derived moduli space is smooth and finite-dimensional if the critical locus is compact, Y is a Kahler and carries a non-vanishing holomorphic volume element. A part of the derived moduli space can be interpreted as moduli of holomorphic deformations of the pair (Y,W), but the whole derived moduli space does not have a direct holomorphic interpretation. Such spaces are of great interests as they carry a weakened form of the Frobenius manifold structure, and are expected to describe genus 0 Gromov-Witten invariants for general symplectic manifolds. I'll talk about Fukaya-theoretic and Hodge-theoretic aspects of non-holomorphically deformed LG models, including a generalization of theory of spectra for isolated singularities (joint work in progress with D.Auroux and L.Katzarkov).  Video

Mina Aganagic, UC Berkeley: Homological Link Invariants from Floer Theory

A new relation between homological mirror symmetry and representation theory solves the knot categorification problem.  The symplectic side geometry side of mirror symmetry is a theory which generalizes Heegard-Floer theory from gl(1|1) to arbitrary simple Lie (super) algebras. The corresponding category of A-branes has many special features, which render it solvable explicitly. In this talk, I will describe how the theory is solved, and how homological link invariants arise from it.  Video

Shaoyun Bai, SCGP & Columbia: Twisted Shklyarov Pairing and Applications

Shklyarov pairing plays the role of Poincare pairing on the Hochschild homology of a homologically smooth and proper A infinity category. I will discuss about a twisted version of such a pairing which is an open string analogue of the Poincare pairing on fixed point Floer cohomology. Applications to Lefschetz fibrations, mirror symmetry, and some problems of classical flavor will also be presented. This talk reports joint work in progress with Paul Seidel.  Video  Slides

Sergei Gukov, California Institute of Technology: Coulomb Branches and Rozansky-Witten Theory 

Three-dimensional Rozansky-Witten theory admits a rigorous mathematical definition, due to Kontsevich and Kapranov, and involves many familiar ingredients from the B-model side of the homological mirror symmetry. Starting with a gentle introduction --- that assumes no prior knowledge of the subject --- we will proceed to modern developments that involve non-compact or infinite-dimensional target spaces, such as the cotangent bundle of the affine Grassmannian. Besides the classical works from the 90s and early 2000s, the material is based on joint works with Hiraku Nakajima, Du Pei, and others https://arxiv.org/abs/1809.04638 https://arxiv.org/abs/2005.05347
as well as the current ongoing work.  Video  Slides

Hülya Argüz, University of Georgia: Quivers and Curves in Higher Dimensions

Let Q be a quiver with n vertices. The combinatorics of mutations of Q leads to the construction of an n-dimensional Poisson X-cluster variety U_Q. On the other hand, for any choice of potential W, the representation theory of (Q,W) leads to the construction of a 3-dimensional Calabi-Yau category C_(Q,W). We prove, under certain assumptions, a correspondence between Donaldson-Thomas invariants of C_(Q,W) and counts of rational curves in normal crossings compactifications of U_Q, corresponding to log Gromov--Witten invariants. This is joint work in progress with Pierrick Bousseau.  Video  Slides

Maxim Kontsevich, Institut des Hautes Etudes Scientifiques: Calculations with Getzler-Gauss-Manin Connections for Non-commutative Algebras

Let A be a noncommutative algebra, and a be an element of A. Then for any number t we have a new algebra obtained form A by inverting the element (1-ta). Then we have a Getzler-Gauss-Manin connection on the bundle of periodic cyclic homology of these algebras depending on t. I'll speak about several meaningful examples including quantum tori, (micro)-differential operators and noncommutative Laurent polynomials.  Video

Sergei Gukov, California Institute of Technology: Coulomb Branches and Rozansky-Witten Theory 

Three-dimensional Rozansky-Witten theory admits a rigorous mathematical definition, due to Kontsevich and Kapranov, and involves many familiar ingredients from the B-model side of the homological mirror symmetry. Starting with a gentle introduction --- that assumes no prior knowledge of the subject --- we will proceed to modern developments that involve non-compact or infinite-dimensional target spaces, such as the cotangent bundle of the affine Grassmannian. Besides the classical works from the 90s and early 2000s, the material is based on joint works with Hiraku Nakajima, Du Pei, and others https://arxiv.org/abs/1809.04638 https://arxiv.org/abs/2005.05347
as well as the current ongoing work.  Video  Slides

Pierrick Bousseau, University of Georgia: DT Invariants and Holomorphic Curves

Kontsevich and Soibelman suggested a correspondence between Donaldson-Thomas invariants of Calabi-Yau 3-folds and holomorphic curves in complex integrable systems. After reviewing this general expectation, I will present a concrete example related to mirror symmetry for the local projective plane (partly joint work with Descombes, Le Floch, Pioline), along with applications in enumerative geometry (partly joint work with Fan, Guo, Wu). I will end by an “explanation” of the general correspondence based on the physics of N=2 4d quantum field theories and holomorphic Floer theory.  Video

Tony Yue Yu, California Institute of Technology: Moduli Space of Non-Archimedean Holomorphic Disks

I will describe the moduli space of non-archimedean holomorphic disks in affine log Calabi-Yau varieties, which is foundational to the non-archimedean mirror symmetry program. I will discuss boundary conditions, smoothness, dimension and properness. Smoothness relies on the non-archimedean deformation theory joint with M. Porta. Properness relies on formal models and Temkin’s theory of reduction of germs. Work in progress with S. Keel.  Video

Tudor Padurariu, Columbia University: Categorical Donaldson-Thomas Theory of C^3 and Beyond

Donaldson-Thomas invariants (DT) are (integer) virtual count of sheaves on a threefold. For Calabi-Yau threefold, there are several refinements of DT invariants, for example to a graded vector space whose Euler characteristic is the numerical DT invariant. When the threefold is a local surface, there are further refinements to a dg category, due to Yukinobu Toda. I will explain joint results with Yukinobu Toda on the structure of these categorifications of DT invariants. I will focus on the example of points on C^3. I will also discuss the construction of a categorical analogue of BPS invariants of C^3 and their application in a categorical DT/ Pandharipande-Thomas (PT) correspondence.  Video

Mohammed Abouzaid, Columbia & Stanford Universities: Floer Homotopy Theory

I will discuss recent developments in the construction of Floer generalised homology groups as well as applications. I will also mention work in progress on understanding curvature in this setting.  Video

Tony Yue Yu, California Institute of Technology: Moduli Space of Non-Archimedean Holomorphic Disks

I will describe the moduli space of non-archimedean holomorphic disks in affine log Calabi-Yau varieties, which is foundational to the non-archimedean mirror symmetry program. I will discuss boundary conditions, smoothness, dimension and properness. Smoothness relies on the non-archimedean deformation theory joint with M. Porta. Properness relies on formal models and Temkin’s theory of reduction of germs. Work in progress with S. Keel.  Video

Nikita Nekrasov, Simons Center, Stony Brook: Lax Operators of Integrable Systems from Intersection Theory

Classical (and quantum) integrable systems of Garnier (Gaudin) and elliptic Calogero-Moser type have been long expected to capture the Seiberg-Witten geometry of A-type quiver N=2 gauge theories in four dimensions. The traditional approach to integrable systems casts them in the Lax form. We present the construction of the associated Lax operators from the intersection theory on the moduli space of framed parabolic sheaves on the projective plane. In gauge theory terms, this is related to the studies of surface defects in (Omega-deformed) linear or cyclic quiver N=2 theory. Based on the works with I.Krichever and A.Grekov.  Video

John Pardon, Simons Center, Stony Brook: Counting Curves on Calabi--Yau Threefolds

We introduce the Grothendieck group of Calabi--Yau 1-cycles on almost complex threefolds, which is the "universal" way of counting curves in Calabi--Yau threefolds.  Recent work of Ionel--Parker and Doan--Ionel--Walpuski on the Gopakumar--Vafa conjecture can be viewed as a calculation of this group.  We explain how to adapt everything to the setting of complex (as opposed to almost complex) threefolds.  A corollary of this discussion is that the MNOP conjecture relating Gromov--Witten and Donaldson--Thomas/Pandharipande--Thomas invariants of complex Calabi--Yau threefolds follows from the special case of local Calabi--Yau's, where it is known by work of Bryan--Pandharipande and Okounkov--Pandharipande.  This is work in progress.  Video

Mohammed Abouzaid, Columbia & Stanford Universities: Floer Homotopy Theory

I will discuss recent developments in the construction of Floer generalised homology groups as well as applications. I will also mention work in progress on understanding curvature in this setting.  Video

Atsushi Takahashi, Osaka University: Frobenius Manifolds for Generalized Root Systems of Type D

Motivated by the geometry of Milnor fiber and vanishing cycles, Kyoji Saito introduced the notion of generalized root systems. A generalized root system consists of a lattice, a set of roots and an element of the Weyl group (called a Coxeter element). Due to the choice of a Coxeter element, the notion is finer than the usual one. There are indeed several "finite generalized root systems of type D", however, little was known for the “non-standard ones” due to lack of their geometric construction. From a Laurent polynomial in one variable (with an involution), we deduce a finite generalized root system of type D and construct a "natural" Frobenius manifold whose Frobenius potential is a rational function, which support the Dubrovin's conjecture on algebraic Frobenius manifolds. This is a joint work with Akishi Ikeda, Takumi Otani and Yuuki Shiraishi.  Slides  Video

Mohammed Abouzaid, Columbia & Stanford Universities: Floer Homotopy Theory

I will discuss recent developments in the construction of Floer generalised homology groups as well as applications. I will also mention work in progress on understanding curvature in this setting.  Video

Ernesto Lupercio, CINVESTAV: Progress in Quantum Toric Geometry

In this presentation, I will provide an overview of the latest advancements in the field of quantum topic geometry and the theory of moduli spaces of toric varieties. Additionally, I will delve into some of the conjectural connections to birational geometry, drawing upon the work of Katzarkov, Meersseman, Verjovsky, Boivin, and Lee.  Video

Chiu-Chu Melissa Liu, Columbia University: Higgs-Coulomb Correspondence and Wall-Crossing in Abelian GLSMs

The input data of a gauged linear sigma model (GLSM) consist of a GIT quotient of a complex vector space V by the linear action of a reductive algebraic group G (the gauge group) and a G-invariant polynomial function on V (the superpotential) which is quasi-homogeneous with respect to a C^* action (R symmetries) on V. The Higgs-Coulomb correspondence relates GLSM invariants which are virtual counts of Landau-Ginzburg quasimaps (Higgs branch) and Mellin-Barnes type integrals (Coulomb branch). In this talk, I will describe the correspondence when G is an algebraic torus, and explain how to use the correspondence to study dependence of GLSM invariants on the GIT stability condition. This is based on joint work with Konstantin Aleshkin. Slides  Video

Bernd Siebert, University of Texas: Toward the Logarithmic Hilbert Scheme

Logarithmic geometry provides tools to work relative a normal crossings divisor, including normal crossings degenerations. I will report on work in progress with Mattia Talpo and Richard Thomas to define a natural logarithmic analogue of the ordinary Hilbert scheme. Immediate applications include induced good degenerations of Hilbert schemes of points. Our point of view also suggests a definition of tropical Hilbert schemes. One larger aim is to develop robust logarithmic methods to deal with coherent sheaves in maximal degenerations as they appear in mirror symmetry.  Video

Ilia Zharkov, Kansas State University: Lagrangian SYZ Fibrations

Given an integral affine manifold B with (semi-)simple singularities D in codimension 2, I will explain how to build a symplectic manifold X and its Lagrangian torus fibration X \to B which extends the tautological cotangent torus bundle over B\D. This involves building local models x..z=1+w_1+..w_n  (the Liouville skeleton is a subject in itself) and then gluing, both steps are interesting. This is a joint project with CY. Mak, D. Matessi and H. Ruddat.  Video