Fukaya Categories: Coefficients, Skeleta, Stability Conditions, IHES, July 2-7, 2018

This workshop will take place at IHES (Bures-sur-Yvette, France) on July 2-7, 2018.
The workshop is for invited participants only.
Organizers: Maxim Kontsevich, Ludmil Katzarkov, Tony Pantev, with assistance from
Élisabeth Jasserand and IHÉS staff.


Monday July 2

10:00-11:00 M. Kontsevich: Introduction to stability program
11:30-12:30 C. Simpson: Towards a definition of the Fukaya-Seidel category with coefficients over the complex plane
Abstract: In this segment of our joint work in progress with Haiden, Katzarkov and Pandit, we consider a graph in the plane. We propose a definition of the A-infinity category of objects over the graph, with coefficients in a fixed appropriate DG-category, including disk corrections corresponding to the regions in the plane cut out by the graph. We can write down an explicit set of A-infinity operations, and have a heuristic comparison with Abouzaid’s definition that suggests why the A-infinity axioms should hold.
2:00-3:00 K. Fukaya: A note on the associativity of multiplicative structure of quantum product of divisor complement    (Video)
3:30-4:30 S. Ganatra: Structural results in wrapped Floer theory

Tuesday July 3

10:00-11:00 M. Kontsevich: Fukaya category based on SFT    (Video)
11:30-12:30 M. Kapranov: Perverse schobers, Fukaya categories with coefficients and the “algebra of the infrared”    (Video)
2:00-3:00 M. Abouzaid: HMS for SYZ singularities via immersed Lagrangians    (Video)
3:30-4:30 D. Auroux: Non-Kahler mirror symmetry: the Hopf surface    (Video)
4:45-5:45 T. Yue Yu: Enumeration of non-archimedean curves in higher dimensional log Calabi-Yau varieties    (Video)
Abstract: I will discuss the enumeration of non-archimedean curves in higher dimensional affine log Calabi-Yau varieties containing an open algebraic torus, part of my joint work with S. Keel. This generalizes the previously studied two-dimensional case, and includes cluster varieties arising from representation theory. Many new ideas are developed in order to go beyond the two-dimensional case. In my talk, I will explain various properties of the moduli spaces which lead to the enumeration. Moreover, I will introduce a new notion of “skeletal curves”, curves whose skeleton lies in the essential skeleton of the ambient log Calabi-Yau variety. Such curves play a special role in the theory.
6:00 Buffet

Wednesday July 4

10:00-11:00 M. Kontsevich: Planar case: harmonic paths in buildings    (Video)
11:15-12:15 F. Haiden: Spectral networks and stability conditions: finite and tame examples    (Video)
Abstract: Spectral networks were introduced in theoretical physics in order to count BPS states. They are graphs with additional algebraic data drawn on a surface and minimizing a weighted total length. Heuristically, they are degenerations of higher dimensional special Lagrangian submanifolds. The main mathematical conjecture about them is that they correspond to stable objects of a stability condition on a Fukaya-type category. I will illustrate how this works in some neat examples related to Dynkin quivers and stacky projective lines. This is an ongoing joint project with L. Katzarkov, M. Kontsevich, P. Pandit, and C. Simpson.
12:30-1:30 P. Pandit: Toward a categorical Donaldson-Uhlenbeck-Yau correspondence    (Video)

Thursday July 5

10:00-11:00 T. Ekholm: Chekanov-Eliashberg dg-algebras for Weinstein domains in contact manifolds    (Video)
Abstract: Let W be a Weinstein manifold, then the product of W and the real line is naturally a contact manifold. We consider contact embeddings of a small neighborhood of a W-slice in this manifold and construct the Chekanov-Eliashberg dg-algebra of such embeddings. When W is a cotangent neighborhood of a smooth Legendrian the definition agrees with the Chekanov-Eliashberg algebra of the smooth Legendrian with coefficients in chains in the based loop space. The construction gives gluing formulas for Fukaya categories.
11:30-12:30 V. Shende: Axiomatics of the wrapped Fukaya category    (Video)
2:00-3:00 A. Bondal: Schobers on orbifold Riemann surfaces    (Video)
3:30-4:30 D. Kaledin: Enhancement for categories    (Video)

Friday July 6

10:00-11:00 E. Lupercio: Self-organized criticality and tropical geometry    (Video)
Abstract: In this talk, I describe our joint work with Kalinin, Guzmán-Saenz, Prieto, Shkolnikov, and Kalinina involving the relation of systems in physics with Self-organized criticality (like earthquakes) and the mathematics of tropical geometry, in particular, we construct a continuous dynamical system in tropical geometry that exhibits self-organized criticality.
11:30-12:30 A. Efimov: On the localizing invariants of derived categories of sheaves of modules    (Video)
2:00-3:00 C. Brav: Deformations of Calabi-Yau categories and Poisson brackets of functions    (Video)
Abstract: We explicate two deformation problems for a smooth Calabi-Yau category C, showing in particular that the complexes underlying the Lie algebras controlling these deformation problems are shifts of negative cyclic and of cylic chains of C. Using these results, we show that the natural map from cyclic chains of C to functions on the `moduli of objects’ M_C is a map of (shifted) Lie algebras with respect to the deformation-theoretic Lie structure on cyclic chains and the Poisson bracket on functions induced by the Calabi-Yau structure on C. Our results give a chain level generalisations of classical constructions of Goldman for moduli of local systems and of Hitchin for GL(n)-Higgs bundles. This is joint work with Nick Rozenblyum.
3:30-4:30 P. Horja: Toric Schobers and D-modules    (Video)

Saturday July 7

10:00-11:00 G. Dimitrov: Non-commutative curve-counting invariants    (Video)
11:15-12:15 A. Harder: Filtrations on cohomology and mirror symmetry    (Video)
12:30-1:30 Z. Sylvan: Doubling stops and spherical swaps    (Video)

Welcome coffee will be available from 9:30 each morning Monday-Friday, as well as a buffet lunch after the last morning talk. (No coffee or lunch on Saturday).

List of participants

Mohammed Abouzaid
Denis Auroux
Netanel Blaier
Jonathan Block
Alexey Bondal
Chris Brav
Patrick Clarke
Colin Diemer
George Dimitrov
Ron Donagi
Alexander Efimov
Tobias Ekholm
Yu-Wei Fan
Kenji Fukaya
Sheel Ganatra
Lino Grama
Fabian Haiden
Andrew Harder
R. Paul Horja
A. Hozie
Dmitry Kaledin
Mikhail Kapranov
Ludmil Katzarkov
Gabriel Kerr
Maxim Kontsevich

Yankı Lekili
Yijia Liu
Ernesto Lupercio
Dmitry Orlov
Pranav Pandit
Tony Pantev
Alexander Petkov
Mauro Porta
Vadim Schechtman
Vivek Shende
Artan Sheshmani
Carlos Simpson
Yan Soibelman
Leonardo Soriani Alves
Ted Spaide
Zack Sylvan
Ivan Yakovlev
Tony Yue Yu