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