JHTP Workshop

A-structures in geometry and representation theory

Dates

04.12.2017–08.12.2017

Location

Hausdorff Research Institute for Mathematics (HIM)
Rheinische Friedrich-Wilhelms-Universität Bonn
Poppelsdorfer Allee 45
53115 Bonn
Germany


The lectures will take place at the HIM lecture hall (see the map below).

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..


The workshop is part of the scientific program of the Junior Hausdorff Trimester Program: Symplectic Geometry and Representation Theory (see also the workshop’s page at the HIM’s website).


The following activities are also part of the same scientific program:

This workshop is organised by Sam Gunningham and Travis Schedler.

Both of these events are organised by Anne-Laure Thiel and Daniel Tubbenhauer.

Organisers

Gustavo Jasso (Rheinische Friedrich-Wilhelms-Universität Bonn)

Julian Külshammer (Universität Stuttgart)

Administrative support

Silke Steinert-Berndt (Hausdorff Research Institute for Mathematics)


Schedule
Timetable
  Mo 04.12 Do 05.12 Mi 06.12 Di 07.12 Fr 08.12
9:30 - 10:30 Registration Shklyarov Barbieri Mak Casals
10:30 - 11:00 Coffee Coffee Coffee Coffee Coffee
11:00 - 12:00 Bocklandt Bodzenta Sibilla Keating Egas Santander
12:00 - 15:00 Lunch Lunch Lunch Lunch Lunch
15:00 - 16:00 Lee Diogo Free Walde Ruddat
16:00 - 16:30 Tea & Cake Tea & Cake Free Tea & Cake Tea & Cake
16:30 - 17:30 Rizzardo Pomerleano Free McBreen Free

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

Excursion: On the free afternoon on Wednesday, we plan to visit the Miró exhibit at the Max Ernst Museum Brühl des LVR in Brühl (20 minutes trip from Bonn Hbf). There is also a nice castle in Brühl (the museum and the castle are next to each other) which is also possible to visit. Further details will be given during the workshop.

Speakers Lino Amorim(Kansas State University)

Anna Barbieri(University of Sheffield)
Raf Bocklandt(University of Amsterdam)
Agnieszka Bodzenta(University of Warsaw)
Roger Casals(Massachusetts Institute of Technology)
Luís Diogo(Uppsala University)
Daniela Egas Santander(Freie Universität Berlin)
Ailsa Keating(University of Cambridge)
Sangwook Lee(Korea Institute for Advanced Study)
Cheuk Yu Mak(Institute of Advanced Studies)
Michael McBreen(Massachusetts Institute of Technology)
Daniel Pomerleano(Imperial College London)
Alice Rizzardo(University of Liverpool)
Helge Ruddat(Johannes Gutenberg Universität Mainz)
Dmytro Shklyarov(Technische Universität Chemnitz)
Nicolò Sibilla(University of Kent)
Tashi Walde(Rheinische Friedrich-Wilhelms-Universität Bonn)

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 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)

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.
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Agnieszka Bodzenta (University of Warsaw)

Title: Categorifying non-commutative deformation theory

Abstract: 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)

Title: TBA

Abstract: TBA
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Luís Diogo (Uppsala University)

Title: Monotone Lagrangians in cotangent bundles of spheres

Abstract: 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)

Title: Derived A-infinity algebras and their homotopies

Abstract: 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)

Title: Structures associated to two-variable singularities

Abstract: 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)

Title: Mirror symmetry between Calabi-Yau categories

Abstract: 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)

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, 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)

Title: Mirror Symmetry and exotic t-structures

Abstract: 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)

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 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)

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.
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Helge Ruddat (Johannes Gutenberg Universität Mainz)

Title: Factoring multiplicities of tropical curves via an L-infinity structure on polyvector fields

Abstract: 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)

Title: Matrix factorizations as D-branes

Abstract: 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)

Title: The topological Fukaya category and mirror symmetry for toric Calabi-Yau threefolds

Abstract: 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 (Rheinische Friedrich-Wilhelms-Universität Bonn)

Title: 2-Segal spaces as invertible infinity-operads

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 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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Registration, funding, and accommodation
Registration

Registration is closed since the capacity of the lecture hall has been reached.

Due to limited seating capacity in the lecture hall, those who are interested in participating should send an e-mail to the organisers including their name, e-mail address, and affiliation, as well as a (brief) letter of intent.
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.