Jenny August (Glasgow)
Title: Contraction Algebras: Tilting and Stability
Abstract: Contraction algebras are a class of finite dimensional algebras arising as invariants of 3-fold flops. In this talk, I will discuss their remarkable tilting theory, describing how paths in an associated hyperplane arrangement can be thought of as a complete description of all their tilting complexes. Time permitting, I will then explain how these results further allow us to control the continuous analogue of tilting complexes, namely stability conditions.
Karin Baur (Graz/Leeds)
Title: CM modules from Grassmannians
Abstract: We study the category of Cohen-Macaulay modules over a quotient of a preprojective algebra. We are interested in describing its Auslander-Reiten quiver. In particular, we study tame cases and construct tubes containing rank 1 and rank 2 modules. We relate the number of rigid indecomposable modules with the number of real roots for the corresponding Kac Moody algebra. This is joint work with D. Bogdanic and A. Garcia Elsener.
Raf Bocklandt (Amsterdam)
Title: Towards a different direction in Mirror symmetry
Abstract: Usually mirror symmetry relates an A-model on one space with a B-model on a different space, but it is also interesting to compare both models on the same space. We explore this idea in the case of surfaces and illustrate it with some baby examples.
Tom Bridgeland (Sheffield)
Title: Riemann-Hilbert problems from Donaldson-Thomas theory
Abstract : I’ll talk about a rather experimental programme which aims to use Donaldson-Thomas invariants to define geometric structures on the space of stability conditions of a CY3 triangulated category. I’ll recall the Kontsevich-Soibelman wall-crossing formula and explain why it naturally suggests a class of Riemann-Hilbert problems. These involve piecewise holomorphic maps from the complex plane into an algebraic torus with prescribed discontinuities along a collection of rays. I’ll mostly focus on the case of the A 2 quiver, and explain (following Gaiotto, Moore and Neitzke) how the corresponding Riemann-Hilbert problem can be solved.
Thomas Brüstle (Sherbrooke)
Title: Skew-gentle algebras via orbifolds
Abstract: Skew-gentle algebras are skew group algebras of gentle algebras with an action of order two. Several recent works relate gentle algebras to dissections of marked oriented surfaces, and provide a geometric model for their derived category in terms of curves in the surface. In joint work with Claire Amiot, we use marked surfaces with an automorphism of order two to model skew-gentle algebras and their derived category. We view this as a way towards studying Fukaya categories for orbifolds.
Igor Burban (Paderborn)
Title: Non-commutative nodal curves and derived-tame algebras.
Abstract: Nodal orders are appropriate non-commutative generalizations of the ring k[[x, y]]/(xy). In my talk, based on a joint work with Yuriy Drozd, I am going to discuss the notion of a non-commutative nodal curve and describe the necessary and sufficient conditions of its tameness. Next, I shall introduce a class of derived tame finite-dimensional algebras (including certain gentle, skew-gentle and quasi-gentle algebras) which are derived equivalent to an appropriate non-commutative nodal curve.
Man-Wai Cheung (Harvard)
Title: Tropical disks counting, stability conditions in symplectic geometry and quiver representations
Abstract: Bridgeland developed stability scattering diagrams which then we can discuss scattering diagrams with quiver representations. Scattering diagrams were proposed by Kontsevich-Soibelman, and Gross-Siebert to describe the toric degenerations of Calabi-Yau varieties. The diagrams encode further geometry as well. Together with Travis Mandel, we associate tropical disks counting with quiver representations by using the stability scattering diagrams. Next, together with Yu-Wei Fan and Yu-Shen Lin, we look at the stable objects for the A2 quiver. It is known that the derived Fukaya-Seidel category of the rational elliptic surface is the derived category of the A2 quiver. Thus wish to match the special Lagrangian with the stable objects in the derived category of coherent sheaves.
Yuriy Drozd (Kiev)
Title: Morita equivalence for non-commutative schemes
Abstract: It is a joint work with Igor Burban. A non-commutative scheme X is, by definition, a pair (X, O_X ), where X is a scheme and O_X is a sheaf of O_X-algebras which is quasi-coherent as a O_X-module. We denote by Qcoh X the category of quasi-coherent O_X-modules and by Coh X the category of coherent O_X-modules. We call the non-commutative scheme X noetherian if X is a noetherian scheme and O_X is coherent as an O_X-module.
A quasi-coherent O X -module P is said to be
• locally projective if every point x ∈ X has an affine open neighbourhood U such that P(U ) is a projective O_X(U)-module.
• local generator if every point x ∈ X has an affine open neighbourhood U such that for some n there is an epimorphism of modules nP(U ) → O_X(U).
• local progenerator if it is a locally projective local generator.
Theorem. Let X = (X, O_X) and Y = (Y, O_Y) be noetherian non-commutative schemes.
(1) Let f : X → Y be an isomorphism of schemes and P ∈ Coh X be a local progenerator
such that End_{O_X} (P) is isomorphic to (f ∗ O_Y )^op. Then the functor Φ_P : Qcoh X → Qcoh Y such that Φ_P F = f_∗ Hom_{O_X} (P, F) is an equivalence.
(2) On the contrary, if Φ : Qcoh X → Qcoh Y is an equivalence of categories, there is a unique isomorphism f : X → Y and a unique (up to isomorphism) local progenerator
P ∈ Coh X such that End_{O_X} (P) is isomorphic to (f ∗ O_Y)^op and Φ is isomorphic to Φ_P.
As the functor Φ_P maps coherent modules to coherent ones and any equivalence of the categories Coh X → Coh Y uniquely extends to an equivalence Qcoh X → Qcoh Y, the theorem remains valid if we replace the categories of quasi-coherent modules to those of coherent modules.
The proof is based on the “usual” Morita Theorem and the results of P. Gabriel [1].
[1] Gabriel, P. Des catégories abéliennes. Bull. Soc. Math. France, 90 (1962) 323–448.
Vladimir Fock (Strasbourg)
TBC
Agnès Gadbled (Uppsala)
Title: Categorical action of the braid group of the cylinder: symplectic aspect
Abstract : Khovanov and Seidel gave in 2000 an action of the classical braid group on a category of algebraic nature that categorifies the Burau representation. They proved the faithfulness of this action through the study of curves in a punctured disk (while Burau representation is not faithful for braids with five strands or more). In a recent article with Anne-Laure Thiel and Emmanuel Wagner, we extended this result to the braid group of the cylinder. The work of Khovanov and Seidel also had a symplectic aspect that we now generalize. In this talk, I will explain the strategy and tools to get a symplectic monodromy in our case and prove its injectivity. If time permits, I will explain how this action lifts to a symplectic categorical representation on a Fukaya category that should be related to the algebraic categorical representation. This is a joint work in progress with Anne-Laure Thiel and Emmanuel Wagner.
Mark Gross (Cambridge)
Title: Intrinsic mirror symmetry
Abstract: This is joint work with Bernd Siebert aiming to construct mirror pairs in complete generality, starting with either log Calabi-Yau manifolds (X,D) with maximally degenerate boundary or maximally unipotent degenerations of Calabi-Yau manifolds X->S. We construct the (homogeneous) coordinate ring of the mirror using a new type of logarithmic Gromov-Witten theory of (X,D) called punctured invariants, developed jointly with Abramovich, Chen and Siebert.
Fabian Haiden (Harvard)
Title: Random chain complexes and contact topology
Abstract: I will report on a project which grew out of an attempt to understand the relation between Hall algebras of Fukaya categories on one hand, and certain skein algebras (i.e. links modulo skein-type relations) on the other. Evidence for such a relation has appeared in several recent works. The main construction takes a Legendrian submanifold and assigns to it a ”random object” in a Fukaya category, more precisely an element of the Hall algebra with non-negative coefficients. I will explain the construction, which can be made very algebraic using flags on chain complexes, in detail in the case of Legendrian knots and links.
Ailsa Keating (Cambridge)
Title: HMS for log-Calabi-Yau surfaces
Abstract: Given a log-Calabi-Yau surface (Y, D) with maximal boundary, we will explain how to explicitly construct a mirror Lefschetz fibration w : V → C, and show how to match up derived categories of sheaves associated to the former with Fukaya categories associated with the latter. Joint work with Paul Hacking.
Bernhard Keller (Paris)
Title: On green sequences
Abstract: In this mainly expository talk, I will recall the definition and main properties of maximal green (or reddening) sequences as well as some of their applications. Following Nagao, I will describe the link between green sequences and chains of torsion classes. I will combine this with recent results on torsion classes to obtain a representation-theoretic proof of an important theorem first proved by Greg Muller using scattering diagrams: A maximal green sequence for a quiver induces a maximal green sequence for each full subquiver.
Tyler Kelly (Birmingham)
Title: Open Mirror Symmetry for Landau-Ginzburg Models
Abstract: We will describe what open enumerative geometry and the enumerative B-model look like for the simplest examples of Landau-Ginzburg models (e.g., W = xr and xr + ys) and explain what mirror symmetry means in this context. This is joint work in preparation with Mark Gross and Ran Tessler.
Yankı Lekili (King’s College)
Title: Fukaya categories in higher dimensions and higher Auslander algebras
Abstract: I will make observations relating higher Auslander algebras to partially wrapped Fukaya categories of higher dimensional symplectic manifolds. It turns out that these are building blocks of more complicated Fukaya categories. I will give an example of the more complicated categories which A. Polishchuk and I used to prove that wrapped Fukaya category of the complement of (n+2) generic hyperplanes in CP^n is equivalent to the derived category of x_1x_2...x_(n+1)=0 via constructing categorical resolutions.
Yin Li (King’s College)
Title: Exact Calabi-Yau categories and infinitesimal symmetries
Abstract: For any Weinstein manifold whose wrapped Fukaya category is exact Calabi-Yau, one can construct a distinguished class B in its first degree S^1-equivariant symplectic cohomology. Using this cohomology class, we define a refined intersection number between objects of the compact Fukaya category which are infinitesimally equivariant with respect to B. This generalizes the q-intersection number introduced previously by Seidel-Solomon.
Sebastian Opper (Paderborn)
Title: On auto-equivalences and complete derived invariants of gentle algebras
Abstract: I will talk about results in [2] on the connection between auto-equivalences of a gentle algebra A and diffeomorphisms of its surface S A as considered in [1]. This leads to a derived equivalence classification of gentle algebras and a description of the group of auto-equivalences as an extension of a subgroup of the mapping class group of S A and a group which is described explicitely if A is triangular. If time permits, I will talk about the relationship between spherical twists and Dehn twists and its application to the classification of spherical objects on cycles of projective lines and the classification of orbits of such objects under the action of auto-equivalences.
[1] S. Opper, P.-G. Plamondon, S. Schroll, A geometric model for the derived category of gentle algebras, arXiv:1801.09659 [math.RT] (2018).
[2] S. Opper, On auto-equivalences and complete derived invariants of gentle algebras, arXiv:1904.04859 [math.RT] (2019).
Fan Qin (Shanghai)
Title: Bases for upper cluster algebras and tropical points
Abstract: It is known that many (upper) cluster algebras possess very different good bases which are parametrized by the tropical points of Langlands dual cluster varieties. For any given injective reachable upper cluster algebra, we describe all of its bases parametrized by the tropical points. In addition, we obtain the existence of the generic bases for such upper cluster algebras. Our results apply to many cluster algebras arising from representation theory, including quantized enveloping algebras, quantum affine algebras, double Bruthat cells, etc.
Konstanze Rietsch (King’s College)
Title: Newton-Okounkov bodies for Grassmannians via mirror symmetry
Abstract: In recent work with L. Williams (arXiv:1712.00447) we use a X-cluster structure on the Grassmannian and the combinatorics of planar bicolored graphs to associate a Newton-Okounkov body to each X-cluster. We describe the Newton-Okounkov bodies explicitly using mirror symmetry: we show that they are polytopes whose facets can be read off from A-cluster expansions of the mirror Landau-Ginzburg model. We also give a combinatorial formula for the lattice points of the Newton-Okounkov bodies which has a surprising interpretation in terms of quantum Schubert calculus.
Ivan Smith (Cambridge)
Title: A symplectic view of category O
Abstract: Certain Slodowy slices in Lie theory can be described as affine open subsets of Hilbert schemes of Milnor fibres of simple singularities. This gives a route to understanding their symplectic topology, and the Fukaya-Seidel categories associated to natural Lefschetz fibrations on these slices. For two-block nilpotents, these Fukaya-Seidel categories are related, via work of Khovanov and Stroppel, to parabolic blocks in Bernstein-Gelfand-Gelfand’s category O. This talk describes joint work in progress with Cheuk Yu Mak.
Salvatore Stella (Leicester)
Title: Acyclic cluster algebras via Coxeter double Bruhat cells and generalized minors
Abstract: Cluster algebras come with a canonical partial basis: the cluster monomials. Extending this partial basis to a full basis has been one of the central problems in the theory giving raise to a varied zoo of constructions. In this talk we will explain how Lie theory can be used to relate them.
Specifically, after recalling the basic definitions, we will explain how any acyclic cluster algebra can be seen as the ring of coordinates of a suitable double Bruhat cell in the associated Kac-Moody group. Under this identification we will interpret cluster monomials as generalized minors –certain functions on a Kac-Moody group defined in terms of its representations– and explain how one can use generalized minors to extend cluster monomials to a continuous family of bases of the cluster algebra in the affine cases.
This talk is based on joint works with D. Rupel and H. Williams.
Emmanuel Wagner (Dijon)
Title: Categorification of 1 and of the Alexander polynomial
Abstract: I’ll give a combinatorial and down-to-earth definition of the symmetric gl(1) homology. It is a (non-trivial) link homology which categorifies the trivial link invariant (equal to 1 on every link). Then I’ll explain how to use this construction to categorify the Alexander polynomial. Finally, if time permits, I will relate this construction to the Hochschild homology of Soergel bimodules and speculate about symmetries between symmetric and exterior gl(n) Khovanov-Rozansky link homologies. (joint with L-H. Robert).