### 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
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 DbCoh(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 X0, 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 X0. 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
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
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

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

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)