### Workshop on HMS, University of Miami, January 26-29, 2017

Speakers:

Netanel Blaier

Anthony Blanc

Mandy Cheung

Yoel Groman

Fabian Haiden

Andrew Harder

Justin Hilburn

Hansol Hong

Ailsa Keating

Heather Lee

Pranav Pandit

Mauro Porta

Ted Spaide

Junho Whang

Tony Yu

Jingyu Zhao

**Organizers:** M. Abouzaid, D. Auroux, L. Katzarkov, T. Pantev

This workshop on HMS and related topics will take place at the University of Miami (Coral Gables, FL) on **January 26-29, 2017**.

The format of this year’s event will be different from previous Miami conferences. The focus will be on the work of **postdoctoral fellows** in this area of mathematics. Additionally, a discussion/problem session will be facilitated by Sheel Ganatra and Nick Sheridan.

This event is supported by the Simons Collaboration on Homological Mirror Symmetry.

→ Registration form (Registration is **closed**, the deadline was December 14).

**Venue: ** The conference takes place at the University of Miami Coral Gables campus. Thursday 1/26 through Saturday 1/28, the talks will be held in the **Fieldhouse**. Sunday 1/29 talks will be in **LC190 (Learning Center)**. See campus map.

**Registration fee:** there will be a nominal registration fee, to cover the cost of refreshments (waived for speakers).

**Airport:** Miami International Airport is about 7 miles from campus. The most convenient way to reach the campus is to take a taxi. Participants expecting travel reimbursement: please keep all original receipts. Also remember that reimbursement from NSF funds requires travel to be booked on a US carrier).

### Preliminary Schedule

Thursday January 26

9:30-10:30 | Pranav Pandit: Categorical Kähler Geometry |

11:00-12:00 | Hansol Hong: Flop on A-side and the stability conditions |

1:30-2:30 | Andrew Harder: TBA |

3:00-4:30 | Discussion (moderated by Sheel Ganatra and Nick Sheridan) |

evening | Conference dinner? (to be confirmed) |

Friday January 27

9:30-10:30 | Ailsa Keating: TBA |

11:00-12:00 | Mauro Porta: TBA |

1:30-2:30 | Anthony Blanc: Formal deformations of categories |

3:00-4:00 | Tony Yue Yu: Gluing holomorphic cylinders |

4:30-5:30 | Mandy Cheung: Quiver Grassmannians and scattering diagrams |

Saturday January 28

9:30-10:30 | Fabian Haiden: Refined Harder-Narasimhan filtrations in modular lattices and iterated logarithms |

11:00-12:00 | Jingyu Zhao: Hodge theoretic invariants on equivariant symplectic cohomology |

1:30-2:30 | Heather Lee: TBA |

3:00-4:00 | Junho Whang: Simple loops on surfaces and boundaries of character varieties |

4:30-5:30 | Ted Spaide: TBA |

Sunday January 29

9:30-10:30 | Yoel Groman: TBA |

10:45-11:45 | Netanel Blaier: TBA |

12:00-1:00 | Justin Hilburn: TBA |

### Titles and abstracts

Anthony Blanc: Formal deformations of categories

**Abstract:** I will talk about a joint work with L. Katzarkov and P. Pandit about the deformation theory of k-linear dg-categories for a field k. In general, the Hochschild cohomology complex parametrizes curved A_infty deformations. After the work of Lurie, we show that under boundedness and the existence of a compact generator, every formal deformation is actually uncurved and possesses a compact generator. The proof uses a new description of the space of deformations in terms of !-group actions on the category.

Mandy Cheung: Quiver Grassmannians and scattering diagrams

**Abstract:** Scattering diagrams, theta functions and broken lines were developed in order to describe toric degenerations of Calabi-Yau varieties and construct mirror pairs. Later, Gross-Hacking-Keel-Kontsevich unravel the relation of those objects with cluster algebras. In the talk, we will discuss how we can put representation theory into these objects. We will also see how the broken lines on scattering diagram give a stratification of quiver Grassmannians using this setting.

Fabian Haiden: Refined Harder-Narasimhan filtrations in modular lattices and iterated logarithms

**Abstract:** I will report on joint work with Katzarkov, Kontsevich, and Pandit in which we introduce a canonical refinement of the Harder-Narasimhan filtration which makes sense in the general context of finite length modular lattices. The filtration is naturally labelled by linear combinations of iterated logarithms and in fact has an analytic interpretation related to a minimizing flow on metrized quiver representations. There is a novel wall-crossing phenomenon involving non-linear walls. Conjecturally, the filtration is related to the asymptotics of lagrangian mean curvature flow on the A-side and the heat flow on hermitian bundles on the B-side.

Hansol Hong: Flop on A-side and the stability conditions

**Abstract:** Chan-Pomerleano-Ueda proved HMS between the smoothing X and the crepant resolution Y of the (local) conifold singularity. We study the operation on the Fukaya category of X which is expected to be mirror to the flop on Y.

On the other hand, there is another mirror construction using special Lagrangian spheres in X and their formal deformation theory, which gives a noncommutative resolution of the conifold. I will explain how this noncommutative mirror behaves when we perform the operation on X, and give an interpretation related to stability conditions.

This is a joint work with Y.-W. Fan, S.-C. Lau, and S.-T. Yau.

Pranav Pandit: Categorical Kähler Geometry

**Abstract:** This talk is based on joint work with Haiden, Katzarkov and Kontsevich. I will describe our attempts to formalize and understand the mathematical structures underlying the physical notion of stability for D-branes in string theory. Our work builds on Bridgeland’s notion of stability conditions, and is inspired by ideas from symplectic geometry, non-Archimedean geometry, geometric invariant theory and the Donalson-Uhlenbeck-Yau correspondence.

Junho Whang: Simple loops on surfaces and boundaries of character varieties

**Abstract:** Given a punctured surface, we consider the moduli spaces of special linear rank two local systems with prescribed traces of monodromy around the punctures. We discuss a proof that every such moduli space is log Calabi-Yau in a suitable sense. This property is related to the combinatorics of simple loops on the surface, where an interesting symmetry of generating series emerges.

Tony Yue Yu: Gluing holomorphic cylinders

**Abstract:** I will talk about a gluing formula for counting holomorphic cylinders in log Calabi-Yau surfaces. The formula roughly says that cylinders can be glued together to form longer cylinders, and the number of longer cylinders equals the product of the numbers of shorter cylinders. Our approach uses Berkovich geometry, deformation theory and several ideas from Gromov-Witten theory.

Jingyu Zhao: Hodge theoretic invariants on equivariant symplectic cohomology

**Abstract:** Mirror symmetry was first studied for Calabi-Yau 3-folds. It predicted the genus zero Gromov-Witten invariants on the quintic 3-fold from the variations of Hodge structures on the mirror family. Following Barannikov and Katzarkov-Kontsevich-Pantev, the work of Ganatra-Perutz-Sheridan showed that the Hodge theoretic mirror symmetry can be recovered from the homological mirror symmetry conjecture for Calabi-Yau manifolds. In this talk, we will describe the Hodge theoretic invariants on the equivariant symplectic cohomology obtaining from the homological mirror symmetry for open manifolds.

### List of participants

Mohammed Abouzaid

Lawrence Barrott

Netanel Blaier

Anthony Blanc

Catherine Cannizzo

Nuno Cardoso

Man Wai Cheung

Patrick Clarke

Colin Diemer

Luis Diogo

Yu-Wei Fan

David Favero

Benjamin Gammage

Sheel Ganatra

Elizabeth Gasparim

Michail Gerapetritis

Lino Grama

Yoel Groman

Roberta Guadagni

Fabian Haiden

Andrew Hanlon

Andrew Harder

Jeff Hicks

Justin Hilburn

Hansol Hong

Matei Ionita

Yusuf Baris Kartal

Ludmil Katzarkov

Ailsa Keating

Tyler Kelly

Gabriel Kerr

Oleksandr Kravets

Sukjoo Lee

Heather Lee

Yu-Shen Lin

Timothy Logvinenko

Ernesto Lupercio

Benedict Morrissey

Ignacio Otero

Pranav Pandit

Tony Pantev

Mauro Porta

Nicholas Sheridan

Yan Soibelman

Ted Spaide

Umut Varolgunes

Sara Venkatesh

Junho Peter Whang

Tony Yue Yu

Zhongyi Zhang

Jiingyu Zhao

Peng Zhou

Yan Zhou