WorkshopA_{∞}structures in geometry and representation theory 

Dates 
04.12.2017–08.12.2017 
Location 
Hausdorff Research Institute for Mathematics (HIM)

Description 
This is a focused workshop whose aim is to bring together young researchers interested in the applications of A_{∞}structures to geometry and representation theory as well as in related topics such as mirror symmetry, stability conditions, exceptional collections, etc..
This workshop is organised by Sam Gunningham and Travis Schedler.
Both of these events are organised by AnneLaure Thiel and Daniel Tubbenhauer. 
Organisers 
Gustavo Jasso (Rheinische FriedrichWilhelmsUniversität Bonn) Julian Külshammer (Universität Stuttgart) 
Administrative support 
Silke SteinertBerndt (Hausdorff Research Institute for Mathematics) 
Schedule 

Timetable 
Note: There will be a reception on Monday at 17:30.


Speakers 
Anna Barbieri(University of Sheffield) 

Abstracts 
Anna Barbieri (University of Sheffield) Title: A construction of Frobenius manifold from stability conditions on Rep(Q) Abstract: 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 welldefined 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.
Title: Deforming Dimers Abstract: We discuss various aspects of the deformation theory of dimer models, illustrate these with examples and relate these to mirror symmetry for Riemann surfaces.
Title: Categorifying noncommutative deformation theory Abstract: I will discuss a categorification of the noncommutative deformation theory of n objects in an abelian category. A suitable abelian category plays the role of the noncommutative 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 prorepresentability of the deformation functor. I will also construct a prorepresenting hull for the deformation functor in general situation and discuss how to recover the functor from the hull.
Title: TBA Abstract: TBA
Title: Monotone Lagrangians in cotangent bundles of spheres Abstract: Monotone Lagrangian submanifolds are an important object of study in symplectic topology. We give a Floertheoretic 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.
Title: Derived Ainfinity algebras and their homotopies Abstract: The notion of a derived Ainfinity algebra, introduced by Sagave, is a generalization of the classical Ainfinity algebra, relevant to the case where one works over a commutative ring rather than a field. Special cases of such algebras are Ainfinity 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. Title: Structures associated to twovariable singularities Abstract: This is a report on joint workinprogress with Roger Casals and Jonathan Evans. Given a twovariable isolated hypersurface singularity, we will explain how to obtain an oriented 2complex (or planar graph). Given any oriented twocomplex, 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 quasiisomorphisms; time allowing, we willpresent further speculations on this representation.
Title: Mirror symmetry between CalabiYau categories Abstract: For certain nonCY symplectic manifolds, Kontsevich’s homological mirror symmetry conjecture predicts equivalences between their Fukaya categories(Amodels) and categories of matrix factorizations(Bmodels). The Ainfinity categories on both sides are all equipped with CalabiYau structures(i.e. there are Serre duality pairings which satisfy cyclic symmetry). We show that for some cases(including toric Fano manifolds) the CalabiYau structures on A and Bmodels become equivalent by mirror symmetry. Based on the joint work with Cheolhyun Cho and Hyungseok Shin.
Title: Spherical Lagrangian submanifolds and spherical functors Abstract: 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, KhovanovSeidel, SeidelSmith and SeidelThomas, 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, autoequivaleneces arising from Dehn twist along spherical Lagrangian submanifolds and explain its relations to spherical functors. This is a joint work with Weiwei Wu.
Title: Mirror Symmetry and exotic tstructures Abstract: Characteristic p quantisation allows us to define many unusual tstructures 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 tstructures. No knowledge of modular representation theory will be assumed.
Title: Degenerations from Floer cohomology Abstract: I will explain how, under suitable hypotheses, one can construct a flat degeneration from the symplectic cohomology of log CalabiYau varieties to the StanleyReisner ring on the dual intersection complex of a compactifying divisor. I will explain how this result connects to classical mirror constructions of Batyrev and HoriIqbalVafa as well as ongoing work of Gross and Siebert.
Title: Enhancements in derived and triangulated categories Abstract: 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.
Title: Factoring multiplicities of tropical curves via an Linfinity structure on polyvector fields Abstract: Descendant log GromovWitten 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 Linfinity 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.
Title: Matrix factorizations as Dbranes Abstract: About 15 years ago the physicists Anton Kapustin and Yi Li interpreted matrix factorizations of isolated hypersurface singularities as topological Dbranes in certain topological string models known as the LandauGinzburg models. The talk is devoted to some mathematical aspects and implications of this result. I will start with a review of 2dimensional openclosed topological field theories underlying the LandauGinzburg models and then report on some recent progress towards the problem of constructing topological conformal field theories in the same context.
Title: The topological Fukaya category and mirror symmetry for toric CalabiYau threefolds Abstract: The Fukaya category of open symplectic manifolds is expected to have good localtoglobal properties. Based on this idea several people have developed sheaftheoretic 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 DyckerhoffKapranov, Nadler and SibillaTreumannZaslow. In this talk I will introduce the topological Fukaya category and explain applications to Homological Mirror Symmetry for toric CalabiYau threefolds. This is joint work with James Pascaleff.
Title: 2Segal spaces as invertible infinityoperads Abstract: 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 socalled 2Segal simplicial objects (due to Dyckerhoff and Kapranov) which has applications in algebraic Ktheory 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 infinitycategorical localization of the former. If time permits we briefly discuss the case of cyclic operads and cyclic objects.

Registration, funding, and accommodation 

Registration 
Registration is closed since the capacity of the lecture hall has been reached. 
Funding 
Accommodation expenses for invited speakers will be covered by HIM. Unfortunately we are not able to provide financial support for other participants. 
Accommodation 
Accommodation for the invited speakers will be arranged by the institute. Other participants are expected to arrange their accommodation by themselves. 
Practical information 

Map  
Directions 
See Getting to HIM for detailed information on how to get to the institute. 
WLAN 
Temporary WLAN access information will be provided to registered participants. In addition, eduroam is available everywhere in the institute. 
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