Fukaya Categories: Coefficients, Skeleta, Stability conditions

Marilyn and James Simons Conference Center (IHES)

Marilyn and James Simons Conference Center


35, route de Chartres 91440 Bures-sur-Yvette (France)

List of speakers:

Chris BRAV
Alexander EFIMOV
Tony Yue YU


Organising Committee:



List of participants:

Mohammed Abouzaid
Denis Auroux
Netanel Blaier
Jonathan Block
Alexey Bondal
Chris Brav
Patrick Clarke
Colin Diemer
George Dimitrov
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
Pranav Pandit
Tony Pantev
Alexander Petkov
Mauro Porta
Vivek Shende
Carlos Simpson
Yan Soibelman
Leonardo Soriani Alves
Ted Spaide
Zack Sylvan
Ivan Yakovlev
Tony Yue Yu




Contact: Elisabeth Jasserand
    • 9:30 AM
      Welcome coffee
    • 1
      Introduction to Stability Program
      Speaker: Maxim Kontsevich (IHES)
    • 11:15 AM
      Coffee Break
    • 2
      Towards a definition of the Fukaya-Seidel category with coefficients over the complex plane
      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.
      Speaker: C. Simpson
    • 12:45 PM
    • 3
      A note on the associativity of multiplicative structure of quantum product of divisor complement
      Speaker: K. Fukaya
    • 3:15 PM
      Coffee break
    • 4
      Structural results in wrapped Floer theory
      Speaker: S. Ganatra
    • 9:30 AM
      Welcome coffee
    • 5
      Fukaya category based on SFT
      Speaker: Maxim Kontsevich (IHES)
    • 11:15 AM
      Coffee break
    • 6
      Perverse schobers, Fukaya categories with coefficients and the "algebra of the infrared"
      Speaker: M. Kapranov
    • 12:45 PM
    • 7
      HMS for SYZ singularities via immersed Lagrangians
      Speaker: Mohammed Abouzaid
    • 3:15 PM
      Coffee break
    • 8
      Non-Kahler mirror symmetry: the Hopf surface
      Speaker: Denis Auroux
    • 9
      Enumeration of non-archimedean curves in higher dimensional log Calabi-Yau varieties
      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.
      Speaker: T. Yue Yu
    • 6:00 PM
      Cocktail Party
    • 9:30 AM
      Welcome coffee
    • 10
      Planar case: harmonic paths in buildings
      Speaker: Maxim Kontsevich (IHES)
    • 11:00 AM
      Coffee break
    • 11
      Spectral networks and stability conditions: finite and tame examples
      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.
      Speaker: Fabian Haiden
    • 12
      Toward a categorical Donaldson-Uhlenbeck-Yau correspondence
      Speaker: Pranav Pandit
    • 1:30 PM
    • 9:30 AM
      Welcome coffee
    • 13
      Chekanov-Eliashberg dg-algebras for Weinstein domains in contact manifolds
      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 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.
      Speaker: Tobias Ekholm
    • 11:15 AM
      Coffee break
    • 14
      Axiomatics of the wrapped Fukaya category
      Speaker: Vivek Shende
    • 12:45 PM
    • 15
      Schobers on orbifold Riemann surfaces
      Speaker: Alexey Bondal
    • 3:15 PM
      Coffee break
    • 16
      Enhancement for categories
      Speaker: Dmitry Kaledin
    • 9:30 AM
      Welcome coffee
    • 17
      Self-organized criticality and tropical geometry
      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.
      Speaker: Ernesto Lupercio (Cinvestav - México)
    • 11:15 AM
      Coffee break
    • 18
      On the localizing invariants of derived categories of sheaves of modules
      Speaker: Alexander Efimov
    • 12:45 PM
    • 19
      Deformations of Calabi-Yau categories and Poisson brackets of functions
      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.
      Speaker: Chris Brav
    • 3:15 PM
      Coffee break
    • 20
      Toric Schobers and D-modules
      Speaker: Paul Horja
    • 21
      Non-commutative curve-counting invariants

      In a joint work with L. Katzarkov to data A,P,G,T, where A and T are triangulated categories, we attach a set whose elements are equivalence classes of fully faithful exact functors (from A to T) with property P. The equivalence relation, which will be explained in the talk, depends on G. We are interested in counting and intersection of the elements in this set. In this talk I will give more details and some examples.

      Speaker: G. Dimitrov
    • 22
      Speaker: A. Harder
    • 23
      Doubling stops and spherical swaps
      Speaker: Zack Sylvan