Simons Collaboration on Homological Mirror Symmetry

Mirror Symmetry and Related Topics, Miami, January 28-February 2, 2019

Tentative schedule

Monday January 28

9:30-10:30	M. Kontsevich: Conjectures on stability conditions I: Stability in Fukaya categories    (Video)
11:00-12:00	P. Seidel: Quantum Steenrod operations and formal groups I    (Video)
2:00-3:00	J. Pardon: Floer theory on Liouville sectors I    (Video)
3:30-4:30	Y. Tschinkel: Equivariant arithmetic geometry    (Video)
5:00-6:00	A. Efimov: Homological mirror symmetry for algebraic curves    (Video)

Tuesday January 29

9:30-10:30	P. Seidel: Quantum Steenrod operations and formal groups II    (Video)
11:00-12:00	Y. Soibelman: Fukaya categories and deformation quantization of complex symplectic manifolds    (Video)
2:00-3:00	C. Simpson: Stability and spectral networks in Fukaya categories with coefficients over the disk    (Video)
3:30-4:30	B. Gammage: Mirror symmetry and K-theoretic Coulomb branches    (Video)
5:00-6:00	G. Dimitrov: More finite sets coming from non-commutative counting    (Video)
8pm	Dinner

Wednesday January 30

9:30-10:30	J. Pardon: Floer theory on Liouville sectors II    (Video)
11:00-12:00	M. Kontsevich: Conjectures on stability conditions II: Stability in non-archimedean setting    (Video)
2:00-3:00	P. Seidel: Quantum Steenrod operations and formal groups III    (Video)
3:30-4:30	K. Fukaya: Lagrangian Floer theory of divisor complement and its possible application to gauge theory    (Video)
5:00-6:00	S. Ganatra: The Gamma and Strominger-Yau-Zaslow conjectures: a tropical approach to periods    (Video)

Thursday January 31

9:30-10:30	M. Kontsevich: Conjectures on stability conditions III: Stability in complex geometry and over finite fields    (Video)
11:00-12:00	M. Abouzaid: Virtual fundamental chains    (Video)
2:00-3:00	J. Pardon: Floer theory on Liouville sectors III    (Video)
3:30-4:30	Y. Lekili: A fable on the Fargues-Fontaine curve    (Video)
5:00-6:00	J. Zhao: A big quantum period theorem for toric Fano surfaces    (Video)

Friday February 1

9:30-10:30	B. Siebert: Periods and analyticity of toric degenerations revisited    (Video)
11:00-12:00	P. Pandit: Calabi-Yau structures and spherical functors    (Video)
2:00-3:00	V. Shende: Skeins on branes    (Video)
3:30-4:30	P. Bousseau: Higher genus wall-crossing and quantization    (Video)
5:00-6:00	E. Lupercio: Quantum toric geometry as chimeric geometry    (Video)

Saturday February 2

9:00-10:00	M. Kapranov: Cohomological Hall algebra of a surface and factorization homology    (no video)
10:15-11:15	F. Haiden: Skein = Hall    (Video)
11:30-12:30	D. Kaledin: Hodge theory and Kahler metrics    (Video)

Titles and Abstracts

M. Kontsevich: Conjectures on stability conditions

Lecture 1: Stability in Fukaya categories.

Abstract: I will review an approach by Y.Soibelman and myself to the definition of Fukaya category based on Lagrangian skeletons and Symplectic Field Theory. This approach allows to describe (hypothetically) a class of Bridgeland stability conditions.

Lecture 2: Stability in non-archimedean setting.

Abstract: I will introduce two examples of stability conditions on categories over non-archimedean fields. The first example deals with the extension of stability from the special fiber to the general fiber, formulated in very broad categorical terms. The second example is non-archimedean SYZ fibration, where inspired by but not using mirror symmetry one can define harmonic objects via geometry of microlocal supports of certain sheaves.

Lecture 3: Stability in complex geometry and over finite fields.

Abstract: II’ll formulate convergence conjectures for Fukaya categories and deformation quantizations. Then I’ll concentrate on the moduli stack of derived noncommutative spaces corresponding to (quantized and gerbified) complex tori and the corresponding stability conditions. The hypothetical description uses spinor representations, similar picture holds for supersingular abelian varieties over finite fields.

J. Pardon: Floer theory on Liouville sectors

P. Seidel: Quantum Steenrod operations and formal groups

Abstract: These lectures concern the simplest kind of Gromov-Witten theory (genus zero, on a Fano manifold) as a Cohomological Field Theory on the chain level. One can apply some basic topological ideas to the situation (going back to the early days of cohomology operations). The motivation for revisiting these two classical subjects comes from arithmetic aspects of mirror symmetry, and more specifically, the Picard groups of Fukaya categories.

P. Bousseau: Higher genus wall-crossing and quantization

Abstract: In the SYZ picture of mirror symmetry, singularities of the SYZ fibration produce holomorphic discs which need to be taken into account in the construction of the mirror. The goal of this talk is to say something about the role of the analogue higher genus open curves, in the case of a Calabi-Yau 3-fold product of an holomorphic symplectic surface by a trivial third direction. I will explain that higher genus log Gromov-Witten invariants of log Calabi-Yau surfaces, viewed as limiting algebraic approximations of the open counts, satisfy the q-refined Kontsevich-Soibelman wall-crossing formula. This result can be seen as a particular case of the expected general relation between higher genus open A-model and Chern-Simons theory. I will end by applications to quantization with canonical basis of holomorphic symplectic surfaces and to enumerative geometry of local del Pezzo surfaces.

G. Dimitrov: More finite sets coming from non-commutative counting

Abstract: We begin by defining non-commutative counting invariants. These are, roughly, sets of triangulated subcategories in a given triangulated category and their quotients. Additional structures on these sets, functoriality and concrete examples will be discussed. Finally, I plan to sketch a proof that for any two affine acyclic quivers $Q$, $Q’$ (i.e. quivers of extended Dynkin type) there are only finitely many triangulated subcategories in $D^b(Rep_{K}(Q))$, which are equivalent to $D^b(Rep_{K}(Q’))$, where $K$ is an algebraically closed field.

A. Efimov: Homological mirror symmetry for algebraic curves

Abstract: We will sketch the proof of HMS for algebraic curves: the derived category of a generalized Tate curve of genus >1 over the Novikov field is equivalent to the suitably defined Fukaya category of a trivalent configuration of spheres (the critical locus of the Landau-Ginzburg model). This is joint work with D. Auroux and L. Katzarkov.

K. Fukaya: Lagrangian Floer theory of divisor complement and its possible application to gauge theory

B. Gammage: Mirror symmetry and K-theoretic Coulomb branches

Abstract: Focusing on the example of multiplicative hypertoric varieties, we explain features of homological mirror symmetry for multiplicative Coulomb branches. In these cases the Fukaya category is generated by holomorphic Lagrangian branes transverse to the components of a singular Hitchin fiber, and the mirror BBB-branes can be produced via quantization in characteristic p. If time permits, we will also discuss the case of the Springer resolution. This is based on joint work with Michael McBreen and Ben Webster.

S. Ganatra: The Gamma and Strominger-Yau-Zaslow conjectures: a tropical approach to periods

Abstract: The Gamma conjectures (in one formulation) posit that the mirror symmetry equivalence between de Rham cohomology of a complex variety and quantum cohomology of its symplectic mirror should identify certain integral lattices. While the algebro-geometric lattice consists of integral homology cycles, the symplectic lattice involves twisting the appropriate lattice of cycles by a transcendental multiplicative characteristic class (the “Gamma class”), built from products of Riemann zeta values and Chern classes. We will review this picture and give an explanation for the simplest case of the Gamma conjectures in terms of the SYZ geometric formulation of mirror symmetry, in which zeta values appear as error terms to the “tropical asymptotics” of period integrals. This is joint work with M. Abouzaid, H. Iritani, and N. Sheridan.

F. Haiden: Skein = Hall

Abstract: Recent results of Morton-Samuelson and Cooper-Samuelson have hinted at a general relation between skein algebras and Hall algebras of Fukaya categories of surfaces. We will make this relation precise (at least in the Z-graded case), which involves considering Legendrian links and their associated filtered A-infinity categories. This is closely related to work of Ng-Rutherford-Shende-Sivek which can be interpreted as the special case when the surface is a disk without boundary points. All this should generalize to higher dimensions, though determining a complete set of skein-type relations is currently an open problem.

D. Kaledin: Hodge theory and Kahler metrics

Abstract: For various applications, it would be useful to describe a structure on a DG-algebra A over C that would produce a Kahler metric on the Toen-Vaquié moduli space of A-modules. The prototypical example is coherent sheaves on a projective complex variety, and the Weil-Petersson metric. This has two descriptions: the standard one involving Quillen’s regularized determinants, and the less standard but more general one using Hodge theory and hyperkahler metrics. I want to talk a little bit about this second description, and explain how to get explicit formulas out of it, so that we can compare it to Quillen’s theory and hopefully understand where the regularized determinants come in. This is joint work in progress with Katzarkov, Pandit and Spaide.

M. Kapranov: Cohomological Hall algebra of a surface and factorization homology

Abstract: Let S be a smooth quasi-projective surface over complex numbers. We consider the Borel-Moore homology of the category of coherent sheaves on S with compact support and make it into an algebra using a version of the Hall multiplication. Sheaves with various dimensions of support give various types of Hecke algebras. We study especially the Hecke algebra of 0-dimensional sheaves from the point of view of factorization algebras. Joint work with E. Vasserot.

Y. Lekili: A fable on the Fargues-Fontaine curve

Abstract: We will demonstrate that homogeneous co-ordinate ring of the equal characteristic version of “the curve” of p-adic Hodge theory introduced by Fargues-Fontaine around 2010 can be computed in the mirror as the Dehn twist ring in a Frobenius twisted version of the Fukaya category of the symplectic 2-torus equipped with a locally constant sheaf of rings whose fiber is a perfectoid ring of characteristic p. This is a report on a joint work in progress with David Treumann.

E. Lupercio: Quantum toric geometry as chimeric geometry

Abstract: I will introduce Quantum Toric Geometry, the study of equivariant compactifications of complex quantum tori, and then interpret it in terms of Chimeric Geometry. This is joint work with Katzarkov, Meerssemann and Verjovsky.

P. Pandit: Calabi-Yau structures and spherical functors

Abstract: I will discuss relationships between Calabi-Yau structures on categories, spherical functors, and derived symplectic geometry. This talk is based on joint work with Ludmil Katzarkov and Ted Spaide.

V. Shende: Skeins on branes

Abstract: I will define open Gromov-Witten theory in a CY 3-fold with coefficients in the skein modules of the Lagrangian branes, and prove the Ooguri-Vafa formula relating the HOMFLYPT polynomial of a knot with the count of holomorphic curves in the conifold ending on a Lagrangian obtained from the knot conormal. This is joint work with Tobias Ekholm.

B. Siebert: Periods and analyticity of toric degenerations revisited

Abstract: Toric degenerations provide a flexible technique for studying all kinds of phenomena related to mirror symmetry. The talk reports on a strengthening of the 2014 paper with Helge Ruddat concerning period integrals in toric degenerations related to canonical coordinates in mirror symmetry. The progress concerns the appearance of the complex Ronkin function in local mirror symmetry, cohomological formulations of the period integral, finite determinacy of toroidal singularities and the analyticity of families with simple singularities.

C. Simpson: Stability and spectral networks in Fukaya categories with coefficients over the disk

Abstract: This is a new report on our joint work with Haiden, Katzarkov and Pandit on the construction of spectral network representatives for stable objects in Fukaya categories with coefficients, over the complex disk as a base. We’ll discuss the definition of the category defined over the Novikov ring with its two functors to categories over the Novikov field and residue field. We’ll show the pictures of our classification of spectral networks for the case of A₅ ⊗ A₂.

Y. Soibelman: Fukaya categories and deformation quantization of complex symplectic manifolds

Abstract: I am going to discuss a long standing question about analogies between the Fukaya category and the deformation quantization from the point of view of the project “Holomorphic Floer Theory”, joint with Maxim Kontsevich. In particular I will discuss how one can enlarge the Fukaya category of a complex symplectic manifold by adding “coisotropic A-branes”.

Y. Tschinkel: Equivariant arithmetic geometry

J. Zhao: A big quantum period theorem for toric Fano surfaces

Abstract: We prove a big quantum period theorem which explicitly relates the descendant log Gromov-Witten invariants of the toric Fano surfaces with the oscillatory integrals of the bulk-deformed superpotentials. This recovers a previous result of Gross for P². This work is joint with Hansol Hong and Yu-Shen Lin.