Conference on Fukaya Category and Homological Mirror Symmetry · Bubble tree compactification of...

Conference on Fukaya Category and

Homological Mirror Symmetry

August 15– 20 , 2019

Peking University

Scientific Committee

Mohammed Abouzaid (Columbia University) Kenji Fukaya(Simons Center)

Chiu-Chu Melissa Liu (Columbia University) Kaoru Ono (RIMS)

Organizing Committee

Bohui Chen (Sichuan University),

Huijun Fan (SMS, Peking University),

Bohan Fang (BICMR, Peking University)

Sponsored by

School of Mathematical Scicence, Peking University

Beijing International Center for Mathematical Research, PekingUniversity

NSFC PKU

CONTENTS

01

Timetable

04

Talk

09

List of participants

15

Conference Information

19

General Information

Conference on Fukaya Category and Homological Mirror Symmetry is

sponsored by the School of Mathematical Sciences and BICMR at Peking

University. The aim of this conference is to bring the active researchers

together to exchange ideas and report their latest progress in Fukaya

category and the related topics. We commemorate the name “Fukaya

category” which has already been known for more than 20 years, and

celebrate Professor Kenji Fukaya's 60th birth.

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Lecture room: 107 Room, Natural Science Classroom Building (理科教学楼 107 教室)

Timetable

Wednesday, August 14

14:00-20:00 Registration（Yanshan Hotel 燕山大酒店）

Thursday, August 15

9:00-9:30 Opening Remark

Chair: Huijun Fan

9:30-10:30 Kenji Fukaya

10:30-11:00 Group Photo / Coffee Break

11:00-12:00 Cheol-hyun Cho

12:30 Lunch

Chair: Bohui Chen

14:00-15:00 Yusuf Baris Kartal

15:00-15:30 Coffee Break

15:30-16:30 Guangbo Xu

16:30-16:45 Coffee Break

16:45-17:45 Hansol Hong

18:00 Dinner

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Friday, August 16

Chair: Kenji Fukaya

10:00-11:00 Tobias Ekholm

11:00-11:15 Coffee Break

11:15-12:15 Kazushi Ueda

12:30 Lunch

Chair: Jianxun Hu

14:00-15:00 Bai-ling Wang

15:00-15:30 Coffee Break

15:30-16:30 Dingyu Yang

16:30-16:45 Coffee Break

16:45-17:45 Mohammad Tehrani

18:00 Dinner

Saturday, August 17

Chair: Tobias Ekholm

10:00-11:00 Vivek Shende

11:00-11:15 Coffee Break

11:15-12:15 Yanki Lekili

12:30 Lunch

Chair: Cheol-hyun Cho

14:00-15:00 Penka Georgieva

15:00-15:30 Coffee Break

15:30-16:30 Rui Wang

17:30 Dinner

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Sunday, August 18

10:00-18:00

Free Discussion

18:00 Banquet (Yajing Room,Yanshan Hotel)

Monday, August 19

Chair: Kaoro Ono

10:00-11:00 David Nadler

11:00-11:15 Coffee Break

11:15-12:15 Emanuel Scheidegger

12:30 Lunch

Chair: Mohammed Abouzaid

14:00-15:00 Andrew Hanlon

15:00-15:30 Coffee Break

15:30-16:30 Suguru Ishikawa

17:30 Dinner

Tuesday, August 20

Chair: Bohan Fang

10:00-11:00 Siu-Cheong Lau

11:00-11:15 Coffee Break

11:15-12:15 Mohammed Abouzaid

12:30 Lunch

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Talks

Local Fukaya categories (Mohammed Abouzaid) Abstract: I will explain joint work with Yoel Groman and Umut Varolgunes whose purpose is to define a Fukaya category associated to each compact subset of a closed symplectic manifold. Much of the difficulty has to do with curvature, and I will explain how to build the necessary algebraic toolkit to resolve them. Auslander-Reiten quivers of ADE curve singularities and Lagrangian Floer theory (Cheol-Hyun Cho)

Abstract: For ADE singularity, indecomposable Cohen-Macaulay modules (or equivalently $\Z/2$-graded matrix factorizations) are classified and described by Auslander-Reiten quivers. For ADE curve singularity $f(x,y)$, we consider Berglund-H\"ubsch dual singularity $f^t(x,y)$ and their Milnor fiber together with diagonal symmetry group $G$. We consider $G$-equivariant Lagrangian Floer theory of the Milnor fiber to construct a full functor to matrix factorization category of $f$, And discuss the relationship of AR quiver and Lagrangian Floer theory. This is a joint work with Dongwook Choa and Wonbo Jung. Augmentation, Alexander polynomial, and Annuli (Tobias Ekholm) Abstract: We give a formula for the Alexander polynomial of a knot in terms of its Augmentation polynomial. The formula gives a certain deformation of the Alexander polynomial. Lagrangian Floer theory of Divisor complement (Kenji Fukaya) Abstract: I report a work in progress on the Lagrangian Floer theory of divisor complement.I will explain how the `Myer Vietoris type sequence' of Lagrangian Floer homology is expected to be obtained when family of symplectic manifolds degenerates to a union of two symplectic manifolds which intersect transversally on the smooth divisor. A Klein TQFT : the local real Gromov-Witten theory of curves（Penka Georgieva）

Abstract: The local Gromov-Witten theory of curves studied by Bryan and Pandharipande revealed strong structural results for the local GW invariants, which were later used by Ionel and Parker in the proof of the Gopakumar-Vafa conjecture.

In this talk I will report on a joint work with Eleny Ionel on the extension of these results to the real setting. We show that the local real GW theory gives rise to a 2-dimensional Klein TQFT defined on an extension of the category of unorientable surfaces.

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We use this structure to completely solve the theory by providing a closed formula for the local real GW invariants in terms of representation theoretic data. As a corollary we obtain the local version of the real Gopakumar-Vafa formula. In the case of the resolved conifold the partition function of the real GW invariants agrees with that of the SO/Sp Chern-Simons theory on S^3. Monomial admissibility and monodromy (Andrew Hanlon)

Abstract: We will discuss monomial admissibility conditions for Laurent polynomials and their corresponding Fukaya-Seidel categories. This discussion will include the relationship of this definition of the Fukaya-Seidel category with more traditional definitions. We will use this framework to compute certain monodromy functors and natural transformations in the mirror to a smooth compact toric variety and their relationship with homological mirror symmetry. Noncommutative resolutions of singularities from Lagrangian deformation (Hansol Hong)

Abstract: Given a Lagrangian in a symplectic manifold, one can consider its Maurer-Cartan deformation which produces a local chart of the mirror that encodes mirror geometry near this Lagrangian. Construction applies to the union of Lagrangian spheres in a certain open symplectic manifold, and produces noncommutative resolutions of well-known algebraic singularities, which are in the form of quivers with potentials. In this talk, I will examine such a construction in different dimensions, and explain how quivers can be used to effectively compare mirror geometries. Construction of symplectic field theory and smoothness of Kuranishi structure (Suguru Ishikawa)

Abstract: Symplectic field theory (SFT) is a generalization of Gromov-Witten invariant and Floer homology for contact manifolds and symplectic cobordisms between them, which was introduced by Eliashberg, Givental and Hofer around 2000. Its algebraic structure was well studied by them, but its construction was a difficult problem. Recently, I succeeded in its construction by usign Kuranishi theory. Kuranishi theory is a theory developed by Fukaya and Ono for the construction of Gromov-Witten invariant and Floer homology for general symplectic manifolds. To use this theory for the construction of SFT, we need to construct smooth Kuranishi structures of moduli spaces. In this talk, I will talk about my work on the construction of SFT and smoothness of Kuranishi structure. Distinguishing symplectic manifolds via the continuous dynamics on wrapped Fukaya categories (Yusuf Baris Kartal)

Abstract: This talk is about distinguishing symplectic manifolds by using the differences in amounts of ``vector fields on Fukaya categories that integrate to periodic flows''. More precisely, we use an algebraic incarnation of the classical symplectic invariant- called Flux- on

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deformations of wrapped Fukaya categories to partially classify open symplectic mapping tori. As an application, we obtain pairs of diffeomorphic Weinstein fillings of a contact manifold that cannot be distinguished by their symplectic cohomology groups, but that are different as Liouville domains. T-equivariant disc potentials (Siu-Cheong Lau) Abstract: Lagrangian Floer theory developed by Fukaya-Oh-Ohta-Ono has played a central role in symplectic geometry and mirror symmetry. In particular, we have a well-defined disc potential for weakly unobstructed Lagrangians. In this talk, we will study and compute a torus-equivariant version of the disc potential for Lagrangian tori and certain Lagrangian immersions which are important for the SYZ construction. This is a joint work with Yoosik Kim and Xiao Zheng. Title: A symplectic look at the Fargues-Fontaine curve (Yanki Lekili) Abstract: I will explain how to introduce a Frobenius twist in the construction of Fukaya category. This new construction applied to a symplectic 2-torus gives a symplectic mirror to the Fargues-Fontaine curve of p-adic Hodge theory. This is joint work in progress with David Treumann. Arboreal skeleta (David Nadler)

Abstract: I will discuss progress in understanding Weinstein manifolds via skeleta with simple singularities. Joint work with Y. Eliashberg and D. Alvarez-Gavela.

TBA (Emanuel Scheidegger) Skeins on branes (Vivek Shende)

Abstract: I will describe how to count open higher genus curves in Calabi-Yau 3-folds, and give an enumerative interpretation of the coefficients of the HOMFLYPT polynomial. This is joint work with Tobias Ekholm. Time permitting, I will speculate on categorification.

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Deformation theory of pseudoholomorphic curves relative to an SNC divisor (Mohammad Farajzadeh Tehrani)

Abstract: Moduli spaces of pseudoholomorphic curves in the presence of an SNC (simple normal crossings) divisor appear naturally in Gromov-Witten theory, Mirror Symmetry, and various other applications. In this talk, we introduce an analytical setup for studying the deformation theory of such curves. In particular, we derive an analog of Ruan-Tian perturbations of the Cauchy-Riemann operator for these moduli spaces. Such perturbations, together with a new compactification that was introduced recently, enable a geometric construction of Gromov-Witten type invariants for certain semi- positive pairs in arbitrary genera. Mirror symmetry for Grassmannians and cluster transformations (Kazushi Ueda)

Abstract: We will discuss a construction of Rietsch's mirror for Grassmannians from the point of view of cluster structures, and the description of the Fukaya categories in terms of matrix factorizations. If the time permits, we will also discuss the other direction of homological mirror symmetry. This is a joint work in progress with Yuichi Nohara. Chen’s proof of the Kotshchick-Morgan conjecture and (equivaraint) K-theoretical Donaldson invariants (Bai-Ling Wang) Abstract: In this talk, I will first briefly review two important results in Bohui Chen’s PhD thesis: Bubble tree compactification of instanton moduli spaces and his proof of the Kotshchick-Morgan conjecture. As a new application, we will define the K-theoretical Donaldson invariant and establish its wall crossing formula for b^+ =1 manifolds. In the presence of compact Lie group (such as U(1)) action, there is an equivariant version of these results. This is work in progress with Bohui Chen. On the Hamiltonian Gromov—Witten invariants for compact symplectic manifolds (Rui Wang)

Abstract: Based on joint work with Bohui Chen and Bai-Ling Wang, we explain the construction of Hamiltonian Gromov—Witten invariants for a compact symplectic manifold which admits compact Lie group Hamiltonian action. The invariants are constructed through moduli spaces of symplectic vortices with cylindrical-ended metrics. Analytic setup for the moduli spaces will be explained. Applications, including a version of quantization for Kirwan morphism (under the assumption that the Lie group is abelian) will be introduced. A Compactness Theorem for translation invariant ASD equation and Atiyah--Floer conjecture（Guangbo Xu）

Abstract: In this talk we will consider the anti-self-dual equation over the product of the real line and a three-manifold with cylindrical end for gauge group SO(3). I will explain the proof

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of a Gromov--Uhlenbeck type compactness result for this equation. This is the first step towards constructing a natural bounding cochain for the symplectic side of the SO(3) Atiyah--Floer conjecture. Fukaya category of Landau-Ginzburg model and the algebra of the infrared (Dingyu Yang) Abstract: In the first part of the talk, I will go over my joint work with Huijun Fan and Wenfeng Jiang on how to define a Fukaya category from a tame holomorphic Morse function on a Kähler manifold, where some of the key points are the use of Witten equation and a highly non-trivial C^0 compactness argument. Then, I will explain how to use Witten equation to construct moduli spaces underlying the algebra of infrared of Gaiotto-Moore-Witten, which incorporates our construction as a natural sub-picture. Here, the space of all pointed subdivisions of the convex hull of the critical values of the holomorphic Morse function provides the setting of coherent families of Witten equation solutions. Mathematically, the algebra structure is formulated by Kapranov-Kontsevich-Soibelman as L infinity structure of secondary polytope and its restriction to A infinity structure, and I will outline how to approach some of their conjectures. If time permits, I will touch upon the wall-crossing aspect and mirror symmetry.

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List of Participants Mohammed Abouzaid Columbia University

abouzaid@math.columbia.edu

Zhuo Chen Beijing Normal University

Xiaojun Chen Sichuan University

xjchen@scu.edu.cn

Bohui Chen Sichuan University

bohui@cs.wisc.edu

Cheol-hyun Cho Seoul National University

chocheol@gmail.com

Dongwook Choa Seoul National University

jwadongwook@gmail.com

Jiajun Dai Sichuan University

jjdai@mail.ustc.edu.cn

Hao Ding Southwest Jiaotong University

dinghao@swjtu.edu.cn

Chengyong Du Sichuan Normal University

yd9966@hotmail.com

Tobias Ekholm Uppsala

Tobias.Ekholm@math.uu.se

Huijun Fan Peking University

fanhj@math.Pku.edu.cn

Bohan Fang Peking University

bohanfang@gmail.com

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Kenji Fukaya Simons Center

kfukaya@scgp.stonybrook.edu

Penka Georgieva IMJ-PRG

penka.georgieva@imj-prg.fr

Wenmin Gong Beijing Normal University

11112018046@bnu.edu.cn

Andrew Hanlon Berkeley

a.hanlon@berkeley.edu

Xingbang Hao

xingbanghao@gmail.com Sun Yat-sen University

Hailong He Nanjing Normal University

hailongher@126.com

Yue He Nanjing Normal University

heyue@njnu.edu.cn

Weiqiang He Sun Yat-sen University

hewq@mail2.sysu.edu.cn

Hansol Hong Yonsei University

hansolhong@yonsei.ac.kr

Wenchuan Hu Sichuan University

huwenchuan@gmail.com

Jianxun Hu Sun Yat-sen University

stsjxhu@mail.sysu.edu.cn

Jin Huang Peking University Shenzhen Graduate School

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13528721065@139.com

Kei Irie University of Tokyo

iriek@ms.u-tokyo.ac.jp

Suguru Ishikawa Kyoto University

suguru@kurims.kyoto-u.ac.jp

Wenfeng Jiang Sun Yat-sen University

wen_feng1912@outlook.com

Rongrong Jin Civil Aviation University of China

Yusuf Baris Kartal MIT

kartal@mit.edu

HuaZhong Ke Sun Yat-sen University

kehuazh@mail.sysu.edu.cn

Siu-Cheong Lau Boston University

lau@math.bu.edu

Yanki Lekili King's College

yanki.lekili@kcl.ac.uk

Xiaobin Li Southwest Jiaotong University

lixiaobin@home.swjtu.edu.cn

Aijin Lin National University of Defence Technology

aijinlin@pku.edu.cn

Yijie Lin Sun Yat-sen University

yjlin12@163.com

David Nadler Berkeley

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denadler@gmail.com

Kaoru Ono RIMS

ono@kurims.kyoto-u.ac.jp

Emanuel Scheidegger Peking University

esche@bicmr.pku.edu.cn

Vivek Shende Berkeley

vivek@math.berkeley.edu

Li Sheng Sichuan University

lshengscu@gmail.com

Kun Shi Beijing Normal University

Tao Su ENS-CNRS

taosu@dma.ens.fr

Shanzhong Sun Capital Normal University

sunsz@mail.cnu.edu.cn

Yuhan Sun Stony Brook University

sun.yuhan@stonybrook.edu

Mohammad Tehrani University of Iowa

mohammad.f.tehrani@gmail.com

Kazushi Ueda University of Tokyo

kazushi@ms.u-tokyo.ac.jp

Bai-ling BRYAN Wang ANU

bai-ling.Wang@anu.edu.au

Rui Wang UC-Berkeley

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ruiwang@berkeley.edu

Yi Wang Stony Brook University

yi.wang.10@stonybrook.edu

Gehao Wang Sun Yat-sen University

gehao_wang@hotmail.com

Yunfeng Wang Science and Technology of China

yfwang89@mail.ustc.edu.cn

Maosong Xiang Huazhong University of Science and Technology

2018210323@hust.edu.cn

Guangbo Xu Texas A&M University

gxu@scgp.stonybrook.edu

Dingyu Yang Humboldt-Universität zu Berlin

dingyu_yang@hotmail.com

Chenglang Yang Peking University

yangcl@pku.edu.cn

Mingzhi Yang Sun Yat-sen University

yangmzh8@mail2.sysu.edu.cn

Futoshi Yagi Southwest Jiaotong University

yagi_futoshi@yahoo.co.jp

Sirui Yu Sichuan University

banaenoptera@163.com

Jinheng Zeng Sichuan University

zengjh662@163.com

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Bingyu Zhang Institut Fourier, Universite Grenoble Alpes

zhangbingyu15@mails.ucas.ac.cn

Rui Zhang Huazhong University of Science and Technology

lrzhang@hust.edu.cn

Shuo Zhang Berkeley

felix.zhang@berkeley.edu

Shuji Zhao Tongji University

102011@tongji.edu.cn

Jian Zhou Tsinghua University

jianzhou@mail.tsinghua.edu.cn

Zhengyi Zhou Institute for Advanced Study

zhengyizhounju@gmail.com

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Conference Information

1. Conference Venue

107 Room, Natural Science Classroom Building (理科教学楼 107 教室)

2. Hotel

Beijing Yanshan Hotel （北京燕山大酒店）

Address: A38 Zhong Guan Cun Street,Hai Dian District,Beijing

Check-in Time From 14:00

Check-out Time To 14:00

Tel.: +86 10 62563388

Convenient stores and Coffee Shops are near Yanshan Hotel with 50 meters.

Please check in directly with your passport.

3. Registration Time: 14:00-20:00, August 14th

Venue: Beijing Yanshan Hotel （北京燕山大酒店）

4. Meals

Breakfast

August 15th-20th（6:30-10:00): Beijing Yanshan Hotel

Lunch August 15th-17th, August 19th-20th (12:00-14:00)

Venue: Time Western Restaurant, 2nd Floor, Building No. 2, Zhongguanyuan Global

Village, Peking University（中关新园 2 号楼二层时光西餐厅）

Dinner August 15th-17th, August 19th (17:30-20:00)

Venue: Time Western Restaurant, 2nd Floor, Building No. 2, Zhongguanyuan Global

Village, Peking University（中关新园 2 号楼二层时光西餐厅）

Banquet Time: 18:00-20:00, August 18th

Venue: Yajing Room, Yanshan Hotel

All registered participants are provided with lunch and dinner coupons for 6 days,

Please submit the coupon to the staff of Time Western Restaurant when taking meals.

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5.Traffic information

Transportation from airport to Yanshan Hotel

The most convenient way is to take a taxi at the Beijing International Airport and go directly

the hotel. The taxi fee is about RMB100 to RMB130 depending on the traffic situation. You can

show the map and the Chinese hotel name (燕山大酒店，海淀) to the driver.

If you take the subway, please first take the airport fast line to Sanyuan Bridge Station (三

元桥) . The single way ticket is about RMB 25. Then buy a single way subway ticket and transfer

to line 10 (direction to Sun Palacet 太阳宫) to Haidian Huangzhuang Station （海淀黄庄）and

transfer to line 4 (direction toTian gongyuan 天宫院 ) and get off at Renming University

Station(人民大学) , then walk out from Exit A2 to the ground. Finally it takes 8-10 minute walk

to Yanshan Hotel.

Transportation between Peking University and Yanshan Hotel

Please take the subway line 4 at Renming University Station (direction to Anhe Bridge

North 安河桥北) and get off at Peking University Station (北京大学东门). The Exit gate is A

Yanshan Hotel

Peking University

Renming University Station

East Gate of Peking University Station

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Place of Natural Science Classroom Building and Building No. 2, Zhongguanyuan

Global Village from Exit A of East Gate of Peking University Station

6. Map of Zhongguanyuan Global Village, Peking University

The entrance of Zhongguanyuan Global Village, Peking University, is on the Zhongguancun North Street.

East Gate of Peking University Station

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General Information

1. Emergency Contacts

Dr. FAN Huijun +86-135-5268-6981

Dr. FANG Bohan +86-185-1833-1563

Ms. Chen Pingping +86-184-0162-1500

Ambulance: 120 Police:110

2. Name Badge

For identification purpose, badges are expected to be worn at all times during

the conference.

3. Internet

If you have an eduroam account, you will be able to get internet access in PKU.

Free cable network is available in your room at Beijing Yanshan Hotel

4. Taxi

The minimum charge is RMB13. After 3 kilometers, RMB2.3 is added every

kilometer. The charge will be 20% higher after 15 kilometers or during the

night time (11pm–5am). Please request a receipt from the taxi driver in case

you leave belongings in the taxi.

5. Currency Exchange

Most banks provide exchange service for foreign currency and traveler’s

checks. Credit cards such as Mastercard, Visa, JCB, Diners are accepted in most

hotels, shopping centers and restaurants. However, they may not be accepted at

small shops or restaurants.

6. Tips & Tax

Tip is not expected or commonly practiced in Beijing. Taxes are already

included in the stated prices.

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Map of PKU Main Campus