Schedule of the Workshop: A-infinity structures in geometry and representation theory

Monday, December 4

9:30 - 10:30 Registration
10:30 - 11:00 Coffee break
11:00 - 12:00 Raf Bocklandt (University of Amsterdam): Deforming Dimers
12:00 - 15:00 Lunch break / Discussion
15:00 - 16:00 Sangwook Lee (Korea Institute for Advanced Study): Mirror symmetry between Calabi-Yau categories
16:00 - 16:30 Tea and cake
16:30 - 17:30 Alice Rizzardo (University of Liverpool): Enhancements in derived and triangulated categories
17:30 - Reception

Abstracts

Anna Barbieri (University of Sheffield): A construction of Frobenius manifold from stability conditions on Rep(Q)

In this talk we consider the space Stab(Q) of (Bridgeland) stability conditions on the abelian category of representations of a (suitable) quiver Q. This is a complex manifold, whose geometry is partly governed by the combinatoric of the quiver, and there are well-defined invariants counting semistable objects. We show that, under some assumptions, these data endow Stab(Q) with a structure of Frobenius manifold. I will start by defining a Frobenius manifold and giving some motivations from enumerative geometry, and I will focus on the result for the Dynkin quiver A_n. This is part of a joint work with J.Stoppa and T.Sutherland.

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Raf Bocklandt (University of Amsterdam): Deforming Dimers

We discuss various aspects of the deformation theory of dimer models, illustrate these with examples and relate these to mirror symmetry for Riemann surfaces.

 

Recorded Talk

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Agnieszka Bodzenta (University of Warsaw): Categorifying non-commutative deformation theory

I will discuss a categorification of the non-commutative deformation theory of n objects in an abelian category. A suitable abelian category plays the role of the non-commutative base for a deformation in this approach. A motivation coming from a categorical description of flops will be outlined. I will give sufficient conditions for the pro-representability of the deformation functor. I will also construct a pro-representing hull for the deformation functor in general situation and discuss how to recover the functor from the hull.

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Roger Casals (Massachusetts Institute of Technology): TBA

TBA

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Luís Diogo (Uppsala University): Monotone Lagrangians in cotangent bundles of spheres

Monotone Lagrangian submanifolds are an important object of study in symplectic topology. We give a Floer-theoretic classification of monotone Lagrangians in cotangent bundles of spheres. The argument involves a classification of proper modules over the wrapped Fukaya category. This is joint work with Mohammed Abouzaid.

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Daniela Egas Santander (Freie Universität Berlin): Derived A-infinity algebras and their homotopies

The notion of a derived A-infinity algebra, introduced by Sagave, is a generalization of the classical A-infinity algebra, relevant to the case where one works over a commutative ring rather than a field. Special cases of such algebras are A-infinity algebras and twisted complexes (also known as multicomplexes). We initiate a study of the homotopy theory of these algebras, by introducing a hierarchy of notions of homotopy between their morphisms. In this talk I will define these objects and describe two different interpretations of them as A-infinity algebras in twisted complexes and as A-infinity algebras in split filtered cochain complexes. We use this reinterpretation to show that this hierarchy of homotopies is an extension of the special case of twisted complexes. I will also talk about how this has lead us to the study of model structures in bicomplexes.

This is joint work with Joana Cirici, Muriel Livernet and Sarah Whitehouse

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Ailsa Keating (University of Cambridge): Structures associated to two-variable singularities

This is a report on joint work-in-progress with Roger Casals and Jonathan Evans. Given a two-variable isolated hypersurface singularity, we will explain how to obtain an oriented 2-complex (or planar graph). Given any oriented two-complex, we will then explain how one can associate to it, on the one hand, a group, which in good cases agrees with the fundamental group of the complement of the discriminant locus of a singularity; and on the other hand, various A_infty categories, generalising certain flavours of Fukaya categories of the singularity. We will show how the group acts on some of the categories by quasi-isomorphisms; time allowing, we willpresent further speculations on this representation.

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Sangwook Lee (Korea Institute for Advanced Study): Mirror symmetry between Calabi-Yau categories

For certain nonCY symplectic manifolds, Kontsevich’s homological mirror symmetry conjecture predicts equivalences between their Fukaya categories(A-models) and categories of matrix factorizations(B-models). The A-infinity categories on both sides are all equipped with Calabi-Yau structures(i.e. there are Serre duality pairings which satisfy cyclic symmetry). We show that for some cases(including toric Fano manifolds) the Calabi-Yau structures on A- and B-models become equivalent by mirror symmetry. Based on the joint work with Cheol-hyun Cho and Hyung-seok Shin.

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Cheuk Yu Mak (Institute of Advanced Studies): Spherical Lagrangian submanifolds and spherical functors

Spherical twist is an auto equivalence of a category whose definition is motivated from the Dehn twist along a Lagrangian submanifold inside a symplectic manifold. In the work of Seidel, Khovanov-Seidel, Seidel-Smith and Seidel-Thomas, they discover surprising applications of spherical twists which are related to link invariants, representation theory and algebraic geometry. In this talk, we will discuss a generalization of this story, namely, auto-equivaleneces arising from Dehn twist along spherical Lagrangian submanifolds and explain its relations to spherical functors. This is a joint work with Weiwei Wu.

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Michael McBreen (Massachusetts Institute of Technology): Mirror Symmetry and exotic t-structures

Characteristic p quantisation allows us to define many unusual t-structures on the derived category of coherent sheaves of a hypertoric variety. I will describe joint work with Ben Webster which uses a variant of mirror symmetry to give a transparent geometric interpretation of these t-structures. No knowledge of modular representation theory will be assumed.

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Daniel Pomerleano (Imperial College London): Degenerations from Floer cohomology

I will explain how, under suitable hypotheses, one can construct a flat degeneration from the symplectic cohomology of log Calabi-Yau varieties to the Stanley-Reisner ring on the dual intersection complex of a compactifying divisor. I will explain how this result connects to classical mirror constructions of Batyrev and Hori-Iqbal-Vafa as well as ongoing work of Gross and Siebert.

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Alice Rizzardo (University of Liverpool): Enhancements in derived and triangulated categories

Derived and triangulated categories are a fundamental object of study for many mathematicians, both in geometry and in topology. Their structure is however in many ways insufficient, and usually an enhancement is needed to carry on many important constructions on them. In this talk we will discuss existence and uniqueness of such enhancements for triangulated categories defined over a field.

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Helge Ruddat (Johannes Gutenberg Universität Mainz): Factoring multiplicities of tropical curves via an L-infinity structure on polyvector fields

Descendant log Gromov-Witten invariants of toric varieties match counts of tropical curves weighted by multiplicities that are obtained as indices of maps of lattices by joint work with Travis Mandel. We show how one can express those multiplicities as products of multiplicities of vertices generalizing Mikhalkin’s multiplicity formula. By introducing an L-infinity algebra of logarithmic polyvector fields which extends the tropical vertex group, we prove that iterated brackets in this algebra compute multiplicities. We give applications to scattering diagrams, theta functions, and cluster algebras where the multiplicity formula is particularly nice.

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Dmytro Shklyarov (Technische Universität Chemnitz): Matrix factorizations as D-branes

About 15 years ago the physicists Anton Kapustin and Yi Li interpreted matrix factorizations of isolated hypersurface singularities as topological D-branes in certain topological string models known as the Landau-Ginzburg models. The talk is devoted to some mathematical aspects and implications of this result. I will start with a review of 2-dimensional open-closed topological field theories underlying the Landau-Ginzburg models and then report on some recent progress towards the problem of constructing topological conformal field theories in the same context.

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Nicolò Sibilla (University of Kent): The topological Fukaya category and mirror symmetry for toric Calabi-Yau threefolds

The Fukaya category of open symplectic manifolds is expected to have good local-to-global properties. Based on this idea several people have developed sheaf-theoretic models for the Fukaya category of punctured Riemann surfaces: the name topological Fukaya category appearing in the title refers to the (equivalent) constructions due to Dyckerhoff-Kapranov, Nadler and Sibilla-Treumann-Zaslow. In this talk I will introduce the topological Fukaya category and explain applications to Homological Mirror Symmetry for toric Calabi-Yau threefolds. This is joint work with James Pascaleff.

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Tashi Walde(University of Bonn):2-Segal spaces as invertible infinity-operads

We sketch the theory of (infinity-)operads via Segal dendroidal objects (due to Cisinski, Moerdijk and Weiss). We explain its relationship with the theory of so-called 2-Segal simplicial objects (due to Dyckerhoff and Kapranov) which has applications in algebraic K-theory and in the construction of(categorified) Hall and Hecke algebras. This relationship comes in the form of an explicit functor from the category of trees to the simplex category which exhibits the latter as an infinity-categorical localization of the former. If time permits we briefly discuss the case of cyclic operads and cyclic objects.

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