### 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, E. Lupercio, 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 (Video) |

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

1:30-2:30 | Andrew Harder: Noncommutative projective spaces and perverse schobers (Video) (Slides) |

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

evening | Conference dinner |

Friday January 27

9:30-10:30 | Ailsa Keating: On the symplectomorphism group of some Milnor fibres (Video) |

11:00-12:00 | Mauro Porta: Derived analytic geometry (Video) |

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

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

4:30-5:30 | Mandy Cheung: Quiver Grassmannians and scattering diagrams (Video) (Slides) |

Saturday January 28

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

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

1:30-2:30 | Heather Lee: Homological mirror symmetry for open Riemann surfaces from pair-of-pants decompositions (Video) (Slides) |

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

4:30-5:30 | Ted Spaide: Calabi-Yau structures, spherical functors, and shifted symplectic structures (Video) |

Sunday January 29

9:30-10:30 | Yoel Groman: Wrapped Floer theory and Lagrangian torus fibrations (Video) |

10:45-11:45 | Netanel Blaier: The quantum Johnson homomorphism, and the symplectic mapping class group of 3-folds (Video) (Slides) |

12:00-1:00 | Justin Hilburn: Monopoles, vortices, and Vermas (Video) |

### Titles and abstracts

Netanel Blaier: The quantum Johnson homomorphism, and the symplectic mapping class group of 3-folds

**Abstract:** This talk is about the symplectic isotopy problem: investigating the kernel of the forgetful map $\pi_0 Symp(M,\omega) \to \pi_0 Diff^+(M)$ when $(M,\omega)$ is a symplectic manifold. While many results are known for some time for $dim=4$, very little has been discovered in higher dimensions. Fukaya categories are a great way to organize the entire Floer theory of a manifold into one algebraic gadget, but they are not the ideal tool for every mission. We use a parametrized version of quantum Massey products and some $A_\infty$-deformation theory to introduce a quantum analogue of the Johnson homomorphism (familiar to low-dimensional topologist from the study of the Torelli group). This invariant attaches a ”characteristic class” in Hochschild cohomology to every suitable automorphism. Moreover, in good cases, this class be evaluated by studying the interaction of rational homotopy with Moduli spaces of holomorphic spheres. As a sample application, we consider $X = BL_C \mathbb{P}^3$, the blowup of projective space at a genus four curve. Using ideas from the birational geometry, we construct a symplectomorphism $\phi : X \to X$ with an explicit factorization as a product of six-dimensional Dehn twists. Even though each of the Dehn twists has infinite order in the symplectic mapping class group, we prove that $\phi$ is exotic.

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.

Yoel Groman: Wrapped Floer theory and Lagrangian torus fibrations

**Abstract:** Consider a Lagrangian torus fibration a la SYZ over a non compact base. I will discuss the construction of wrapped Floer theory in this setting. Note that this setting is, in general, not exact even near infinity. I will apply this to homological mirror symmetry in the case where the total space is the complement of an anti-canonical divisor in a toric Calabi Yau.

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.

Andrew Harder: Noncommutative projective spaces and perverse schobers

**Abstract:** I will describe a construction of specific perverse schobers (and generalizations) which will recover well known noncommutative deformations of projective spaces in dimensions 2 and 3. This is joint with L. Katzarkov.

Justin Hilburn: Monopoles, vortices, and Vermas

**Abstract:** Symplectic duality, as described by Braden-Proudfoot-Licata-Webster, is an equivalence of certain categories associated to a pair of conical symplectic singularities. Each such category is a subcategory of modules over a deformation quantization of functions on the corresponding singularity. The prototypical example is when the singularity is the nilpotent cone of a semi-simple Lie algebra g in which case the corresponding category is the Bernstein-Gelfand-Gelfand Category O associated to g.

It is expected that all such dual pairs arise as Higgs and Coulomb branches of 3d N=4 SUSY field theories. By examining a certain twist of this theory one can realize the action of a deformation quantization of the Coulomb branch on a Verma module in terms of the action of monopole operators on the vortex moduli space.

One can then use this to write the partition functions for 2d N=(2,2) theories in terms of Whittaker vectors. This gives a physical formulation and generalization of the finite AGT conjecture which has been studied by Braverman, Finkelberg, Feigin, Frenkel, Kuznetsov, Nakajima, and Rybnikov in the case of a nilpotent orbit closure. This is joint work with Bullimore, Dimofte, Gaiotto, and Kim.

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.

Ailsa Keating: On the symplectomorphism group of some Milnor fibres

**Abstract:** We report on joint work in progress, partly with Jonathan Evans, on symplectic mapping class groups of Milnor fibres of some affine hypersurface singularities. We focus on the image of the fundamental group of the complement of the associated discriminant locus, and on (stabilizations of) two-variable singularities.

Heather Lee: Homological mirror symmetry for open Riemann surfaces from pair-of-pants decompositions

**Abstract:** We will demonstrate one direction of HMS for punctured Riemann surfaces — the wrapped Fukaya category of a punctured Riemann surface is equivalent to the matrix factorization category MF(X,W) of the toric Landau-Ginzburg mirror (X, W). The category MF(X,W) can be constructed from a Cech cover of (X,W) by local affine pieces that are mirrors of pairs of pants. We supply a suitable model for the wrapped Fukaya category for a punctured Riemann surface so that it can also be explicitly computed in a sheaf-theoretic way, from the wrapped Fukaya categories of various pairs of pants in a decomposition. The pieces are glued together in the sense that the restrictions of the wrapped Floer complexes from two adjacent pairs of pants to their adjoining cylindrical piece agree.

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 Donaldson-Uhlenbeck-Yau correspondence.

Mauro Porta: Derived analytic geometry

**Abstract:** I will survey recent developments in derived complex and non-archimedean geometry. After giving some initial motivations, I will move to the main results, that include derived versions of GAGA theorems and derived deformation theory.

I will explain how to use the analytic cotangent complex to prove an analytic version of Artin-Lurie’s representability theorem. I will conclude by giving some applications of this theorems.

Parts of this talk are based on joint work with Tony Yue Yu.

Ted Spaide: Calabi-Yau structures, spherical functors, and shifted symplectic structures

**Abstract:** Calabi-Yau structures—both absolute and relative—are categorical structures that show up on both sides of homological mirror symmetry. They are categorical analogues of shifted symplectic structures in derived algebraic geometry, which show up on many moduli spaces.

Spherical functors are special types of functors which also show up in mirror symmetry. They are connected to the study of perverse schobers.

In this talk I will discuss and give examples of Calabi-Yau structures and shifted symplectic structures, and discuss the connection between spherical functors and Calabi-Yau structures.

This is joint work with L. Katzarkov and P. Pandit.

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

Rodrigo Barbosa

Lawrence Barrott

Netanel Blaier

Anthony Blanc

Catherine Cannizzo

Nuno Cardoso

Man Wai Cheung

Patrick Clarke

Colin Diemer

Luis Diogo

Alexander Efimov

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

John Morgan

Benedict Morrissey

Ignacio Otero

Pranav Pandit

Tony Pantev

Daria Poliakova

Mauro Porta

Larry Richardson

Nicholas Sheridan

Yan Soibelman

Ted Spaide

Umut Varolgunes

Sara Venkatesh

Junho Peter Whang

Tony Yue Yu

Zhongyi Zhang

Jiingyu Zhao

Peng Zhou

Yan Zhou