### Inaugural Conference – University of Pennsylvania, November 5-8, 2015

**Organizers:** M. Abouzaid, D. Auroux, R. Donagi, K. Fukaya, L. Katzarkov, M. Kontsevich, B. Lian, T. Pantev, S.-T. Yau

Speakers:

Mohammed Abouzaid

Kevin Costello

Alexander Efimov

Kenji Fukaya

Sheel Ganatra

Ludmil Katzarkov

Maxim Kontsevich

Siu-Cheong Lau

Bong Lian

David Nadler

Nick Sheridan

Bernd Siebert

Yan Soibelman

Edward Witten

Shing-Tung Yau

Eric Zaslow

The inaugural conference of the Simons Collaboration on Homological Mirror Symmetry took place at the University of Pennsylvania on **November 5-8, 2015**. This event was supported by a grant from the Simons Foundation.

The program included lecture series on the following topics:

- Categorical geometry (M. Kontsevich)
- The SYZ program (S.-T. Yau, S.-C. Lau)
- Family Floer theory (M. Abouzaid, K. Fukaya)
- From HMS to enumerative mirror symmetry (S. Ganatra, N. Sheridan)
- Tropical instanton corrections and log geometry (B. Siebert)
- BCOV quantization (K. Costello)

### Schedule

Thursday November 5 (Arch 108)

8:45-9:00 | Opening remarks (Video) |

9:00-10:00 | M. Kontsevich: Formulation of homological mirror symmetry (Slides) (Video) |

10:15-11:15 | S.-T. Yau: SYZ mirror symmetry conjecture (Slides) (Video) |

11:30-12:30 | N. Sheridan: From categories to curve counts I: automatic split generation and non-degeneracy (Slides) (Video) |

2:00-3:00 | E. Witten: Gauge theory and Khovanov homology (Slides) (Video) |

3:30-4:30 | Y. Soibelman: Riemann-Hilbert correspondence and Fukaya categories (Slides) (Video) |

Friday November 6 (Arch 108)

9:00-10:00 | M. Kontsevich: Kähler stability (Slides) (Video) |

10:15-11:15 | S.-C. Lau: Mirror construction using non-commutative geometry (Slides) (Video) |

11:30-12:30 | S. Ganatra: From categories to curve counts II: variations of semi-infinite Hodge structures (Slides) (Video) |

2:00-3:00 | B. Siebert: Moduli of K3 surfaces and mirror symmetry (Slides) (Video) |

3:30-4:30 | L. Katzarkov: Applications of HMS (Slides) (Video) |

Saturday November 7 (DRL A8)

9:00-10:00 | K. Fukaya: Zero-dimensional family Floer homology (Video) |

10:15-11:15 | E. Zaslow: Knots, Graphs and Clusters (Video) |

11:30-12:30 | K. Costello: An overview of BCOV theory and its quantization I (Video) |

2:00-3:00 | B. Siebert: Toward symplectic cohomology and Lagrangian Floer theory in toric degenerations (Slides) (Video) |

3:30-4:30 | D. Nadler: Applications of arboreal singularities (Video) |

7:00 | Banquet at Penang (117 N 10th St) |

Sunday November 8 (DRL A8)

9:00-10:00 | B. Lian: Periods, residues and mirror symmetry (Slides) (Video) |

10:15-11:15 | A. Efimov: Homological invariants of DG algebras and generalized degeneration conjecture (Video) |

11:30-12:30 | K. Costello: An overview of BCOV theory and its quantization II (Video) |

2:00-3:00 | M. Abouzaid: From family Floer cohomology to homological mirror symmetry (Slides) (Video) |

### Abstracts

K. Costello: An overview of BCOV theory and its quantization

**Abstract:** I’ll discuss the quantum field theory coupling BCOV theory and holomorphic Chern-Simons theory in 3 complex dimensions. I’ll explain what it means to quantize this field theory, and explain how the output of the quantization relates to objects found in the A-model. Finally I’ll sketch the proof of a theorem with Si Li showing that the coupled theory can be quantized in certain circumstances.

A. Efimov: Homological invariants of DG algebras and generalized degeneration conjecture

**Abstract:** We will introduce a new functorial homological invariant for any DG algebra over a directed DG quiver. This invariant comes with a map to the usual Hochschild homology. Conjecturally, this map always vanishes if the basic field has characteristic zero. This conjecture generalizes the degeneration conjecture of Kontsevich and Soibelman.

In the special case when the DG quiver has one vertex we will show that 1) the vanishing holds, and 2) there does not exist an “explicit” proof of this fact.

S. Ganatra: From categories to curve counts 2: variations of semi-infinite Hodge structures

**Abstract:** We will explain how to recover from homological mirror symmetry (HMS) a version of enumerative mirror symmetry, and in particular how to use Sheridan’s proof of HMS for the quintic threefold to give a new proof of enumerative predictions made by Candelas, de la Ossa, Green and Parkes in 1991. The main intermediate step, following ideas of Kontsevich and Barannikov, is extracting from HMS a form of Hodge-theoretic mirror symmetry. The crucial new ingredient on the A side is showing that an open-closed map from the cyclic homology of the Fukaya category to quantum cohomology intertwines certain Hodge-theoretic structures. On the B side, our result relies on a conjecture (some of which is in the literature) that a cyclic HKR map does the same.

S.-C. Lau: Mirror construction using non-commutative geometry

**Abstract:** Strominger-Yau-Zaslow proposed that the mirror of a Kaehler manifold can be constructed from deformations of fibers of a Lagrangian torus fibration. In this talk we develop a construction which is related but different from SYZ. Namely we consider a deformation space of a set of Lagrangian submanifolds which are not necessarily tori. We will see that this space is naturally non-commutative in nature. Moreover it can be regarded as a “mirror” in a generalized sense, namely there exists a natural functor from the Fukaya category to the category of modules over this space. The functor always carries a certain injectivity property. The construction is particularly useful for systems in which collections of Lagrangians are canonically constructed. Calabi-Yau threefolds associated to Hitchin systems over Riemann surfaces provide a beautiful class of examples. This is a joint work with Cheol-Hyun Cho and Hansol Hong.

B. Lian: Periods, residues and mirror symmetry

**Abstract:** Some recent progress on periods of Calabi-Yau manifolds is reviewed. Applications to large complex structure limit problems, generalized hypergeometric functions, periods of general type and Fano varieties are discussed. The talk is based on a series of joint works, some of which in progress, with S. Bloch, J. Chen, A. Huang, D. Srinivas, S.-T. Yau, and X. Zhu.

D. Nadler: Applications of arboreal singularities

**Abstract:** I’ll explain how to calculate Landau-Ginzburg models using Lagrangian skeleta with combinatorial singularities. It’s useful to organize the structure in the form of a perverse schober as introduced by Kapranov-Schechtman. I’ll focus on the example $M=\Bbb{C}^n$, $W=z_1…z_n$ mirror to a linear hypersurface in a toric variety.

N. Sheridan: From categories to curve counts I: automatic split generation and non-degeneracy

**Abstract:** We will explain two structural results in mirror symmetry. The first is about the endgame of proving homological mirror symmetry for a particular Calabi-Yau mirror pair (X,Y). Namely, suppose that you have found full subcategories of Fuk(X) and D^{b}Coh(Y), and proved that they are quasi-equivalent; and on the algebraic geometry side, suppose that the subcategory split-generates, and furthermore that Y has ‘maximally unipotent monodromy’. Our result says that the full subcategory of Fuk(X) then necessarily split-generates too, by Abouzaid’s criterion: so one obtains the full HMS equivalence. The second result is a byproduct of the proof of the first: it says that, under the same hypotheses, if X and Y are homologically mirror, then Fuk(X) is automatically `non-degenerate’ (in the sense that the open-closed map hits the unit); this will be crucial in the second talk in the series. Joint work with Tim Perutz.

B. Siebert: I. Moduli of K3 surfaces and mirror symmetry

**Abstract:** The toric degeneration approach to mirror symmetry seems to add inessential information by insisting on a choice of central fibre rather than working over a non-archimedean field. In the talk I want to convey that it is exactly this property that makes mirror symmetry play well with birational algebraic geometry and toroidal compactifications. The case in point is the modular compactification of moduli spaces of polarized K3 surfaces via mirror symmetry, a program developed by Paul Hacking and Sean Keel in collaboration with Mark Gross (and with minimal contributions from my side).

II. Toward symplectic cohomology and Lagrangian Floer theory in toric degenerations

**Abstract:** For a toric degeneration p:X→D with central fibre X_{0}, the tropical point of view suggests a close connection between the following three entities related to p: (1) Lagrangian Floer theory on the general fibre of p (say, a Calabi-Yau manifold), (2) A version of symplectic cohomology of X, (3) Certain pointed logarithmic Gromov-Witten invariants of X_{0}. The last item opens the possibility of a purely algebraic-geometric description of the Fukaya category in this context. In the talk I will report on the status of this conjecture for the easiest, but already rather interesting case, the degeneration of an elliptic curve (the Tate curve). This is joint work partly with Hülya Argüz and with Mohammed Abouzaid.

Y. Soibelman: Riemann-Hilbert correspondence and Fukaya categories

S.-T. Yau: SYZ mirror symmetry conjecture

E. Zaslow: Knots, Graphs and Clusters

**Abstract:** I will describe the construction of a category based on a Legendrian knot in the co-circle bundle of a surface. It is equivalent to the Fukaya category of the cotangent bundle, with asymptotic conditions defined by the knot. The category is an invariant: Legendrian isotopies induce equivalences. The space of objects in this category is a cluster variety: exact Lagrangian “fillings” of the knot give rise to cluster charts. Given a bipartite graph on a surface, its zig-zag paths determine a Legendrian knot, and one can construct a canonical filling. “Square moves” on the graph correspond to cluster transformations.

Many well-known cluster varieties can be constructed in this way, including wild character varieties and positroid strata of the Grassmannian. I will give simple examples to illustrate these constructions.

This talk is based on joint work with David Treumann, Vivek Shende and Harold Williams. Forthcoming work of Shende-Treumann-Williams shows that for any cluster seed one can engineer a Weinstein manifold whose Fukaya category has a space of objects equal to the corresponding cluster variety.

### List of participants

Leonardo Abbrescia

Mohammed Abouzaid

Mohammad Akhtar

Benjamin Albert

Denis Auroux

Matthew Ballard

Rodrigo Barbosa

Netanel Blaier

Jonathan Block

Catherine Cannizzo

Patrick Clarke

Kevin Costello

James Curtis

Colin Diemer

Ron Donagi

Alexander Efimov

Yu-Wei Fan

Kuan-Ying Fang

Mohammad Farajzadeh Tehrani

David Favero

Teng Fei

Lei Fu

Kenji Fukaya

Sergey Galkin

Sheel Ganatra

Yuan Gao

Peng Gao

Honghao Gao

Michail Gerapetritis

Gary Gibbons

Alexander Goncharov

Yoel Groman

Roberta Guadagni

Fabian Haiden

Daniel Halpern-Leistner

Andrew Hanlon

Jeffrey Hicks

An Huang

Matei Ionita

Xiaofei Jin

Xin Jin

Atsushi Kanazawa

Arjun Kar

Ilya Karzhemanov

Alexander Kasprzyk

Ludmil Katzarkov

Ailsa Keating

Gabriel Kerr

Yoosik Kim

Maxim Kontsevich

Can Kozcaz

Oleksandr Kravets

Enka Lakuriqi

Jonathan Lai

Siu-Cheong Lau

Heather Lee

Sukjoo Lee

Naichung Conan Leung

Hyenho Lho

Bong Lian

Dahisy Lima

Chiu-Chu Melissa Liu

Yuhang Liu

Matthew Mahowald

Cheuk Yu Mak

Michael McBreen

David Nadler

Khoa Nguyen

Pranav Pandit

Tony Pantev

Tim Perutz

Thomas Prince

Hua Qiang

Helge Ruddat

Nick Sheridan

Bernd Siebert

Yan Soibelman

Ran Tessler

Yuri Tschinkel

Umut Varolgunes

Saraswathi Venkatesh

Chin-Lung Wang

Edward Witten

Chris Woodward

Weiwei Wu

Dan Xie

Shing-Tung Yau

Philsang Yoo

Shilin Yu

Eric Zaslow

Jingyu Zhao

Peng Zhou

Yan Zhou

Yuecheng Zhu

### Venue

The conference will take place at the University of Pennsylvania. Thursday and Friday talks will be in room Arch 108 (+ overflow room Arch 109). The Arch building is on the corner of 36th St and Locust Walk. Saturday and Sunday talks will be in Auditorium A8 on the ground floor of David Rittenhouse Laboratory (DRL) (corner of 33rd and Walnut Streets; access through entrance on 33rd St).

**Housing:** most participants who requested a room reservation will be staying at the Club Quarters Hotel, 1628 Chestnut Street, Philadelphia PA 19103. *Please do not contact the hotel directly at this time, as the rooming list has not been finalized.*

The hotel is a 30-to-35 minute walk (1.5 mile) to the conference venue, or you can take SEPTA city buses 21 or 42 running along Walnut St, or go to the subway station on the corner of 15th and Market streets and take either the Market-Frankford (MFL) subway line, westbound, to 34th St, or any of the westbound trolley (green) lines to 33rd or 36th St. (cost: $2.25)

→Map showing hotel and conference venues

→Directions from hotel to ARCH building