# Simons Collaboration on Homological Mirror Symmetry Annual Meeting 2021

Date & Time

Organizer:
Tony Pantev, University of Pennsylvania

Meeting Goals:
The Simons Collaboration on Homological Mirror Symmetry Annual Meeting will focus on recent work which explores arithmetic, topological, and analytic refinements of the homological mirror correspondence. Alongside the foundational work needed to produce such refinements we will discuss some unexpected and exciting applications of the refined mirror constructions. Talks will highlight progress on novel and deeper aspects of the homological mirror correspondence.

On the symplectic side, these include categorical enhancements of mirror maps incorporating directly the dependence of Fukaya categories on the symplectic parameter, recent homotopy theoretic extensions of Floer theory and their implications to Hamiltonian dynamics and the topology of algebraic varieties, and strong finiteness results for symplectic homology that give rise to categorical crepant resolutions and bridge the Gross-Siebert intrinsic mirror symmetry program and the birational non-commutative geometry. On the algebraic side, talks will explore explicit mirror constructions for log and fractional Calabi-Yau varieties.

We will also investigate fundamental questions on the interaction of quantum spectra, quantum blow-up formulas, and semi-orthogonal decompositions of the derived categories of projective varieties. As an application, we will discuss how the semi-continuity of the spectrum combines with the quantum-blow up formula to produce new non-rationality criteria valid in any dimension.

Finally, on the topological side, talks will showcase a recent solution to the knot categorification problem which works uniformly for all gauge groups and is transformed non-trivially by homological mirror symmetry.

Speakers:
Mohammed Abouzaid
Mina Aganagic
Ludmil Katzarkov
Ailsa Keating
Maxim Kontsevich
Tsung-Ju Lee
Daniel Pomerleano
Paul Seidel

• Agenda

#### Thursday, November 18

 8:30 AM CHECK-IN & BREAKFAST 9:30 AM Maxim Kontsevich (Remote) | Blow-up Equivalence 10:30 AM BREAK 11:00 AM Daniel Pomerleano | The Frobenius Property and HMS for log CY Varieties 12:00 PM LUNCH 1:00 PM Ludmil Katzarkov | Spectra and Applications 2:00 PM BREAK 2:30 PM Mina Aganagic | Homological Mirror Symmetry (HMS) and Knot Categorification 3:30 PM BREAK 4:00 PM Paul Seidel (Remote) | Mirror Symmetry and Noncommutative Linear Systems 5:00 PM DAY ONE CONCLUDES

#### Friday, November 19

 8:30 AM CHECK-IN & BREAKFAST 9:30 AM Ailsa Keating | Symplectomorphisms of Some Weinstein Four-Manifolds 10:30 AM BREAK 11:00 AM Tsung-Ju Lee | Mirror Symmetry on Singular Calabi–Yau Spaces 12:00 PM LUNCH 1:00 PM Mohammed Abouzaid | Prospects for Floer Homotopy Theory 2:00 PM MEETING CONCLUDES
• Abstracts

Mohammed Abouzaid
Columbia University

Prospects for Floer Homotopy Theory

Abouzaid will review recent progress in applying Floer homotopy theory to symplectic topology and algebraic geometry, and discuss the interaction of these ideas with homological mirror symmetry.

Mina Aganagic
University of California, Berkeley

Homological Mirror Symmetry (HMS) and Knot Categorification

In this talk, Aganagic will describe an application of homological mirror symmetry (HMS) to solve an old problem from a different branch of mathematics. The problem is to categorify Chern–Simons link invariants in a manner which is uniform with respect to the choice of a Lie algebra and originates from geometry. The solution to the knot categorification problem comes with a new relation between HMS and representation theory. There is a family of mirror pairs (X, Y), labeled by a choice of a simply laced Lie algebra, a collection of its representations, and a weight in their tensor product. Mirror symmetry should be manifest in that the derived category of equivariant coherent sheaves on X and the derived Fukaya–Seidel category of its mirror Y are both equivalent to a derived category of modules of the same associative algebra A, which turns out to be a cousin of the algebra considered by Khovanov and Lauda and by Rouquier. The action of braiding and the branes corresponding to the caps and the caps which close off braids into links all have explicit geometric and algebraic descriptions. The symplectic geometry side of mirror symmetry is a new theory generalizing Heegard–Floer theory. The generalization corresponds to replacing gl(1|1) by an arbitrary simply laced Lie algebra. Extension to nonsimply laced Lie algebras should arise by folding.

Ailsa Keating
University of Cambridge

Symplectomorphisms of Some Weinstein Four-Manifolds

Let $$M$$ be the Weinstein four-manifold mirror to $$Y\backslash D$$ for $$(Y,D)$$ a log Calabi–Yau surface; this is usually the Milnor fiber of a cusp singularity. We introduce two families of symplectomorphisms of $$M$$: Lagrangian translations, which we prove are mirror to tensors with line bundles; and nodal slide recombinations, which we prove are mirror to automorphisms of $$Y$$. Together with spherical twists, these are expected to generate the symplectic mapping class group of $$M$$. Time permitting, some applications will be given. This is based on joint work with Paul Hacking.

Ludmil Katzarkov
University of Miami

Spectra and Applications

Katzarkov will introduce new kinds of spectra related to noncommutative Hodge structures. Applications to birational geometry will be considered.

Maxim Kontsevich
Institut des Hautes Études Scientifiques

Blow-up Equivalence

A couple of years ago, Kontsevich proposed a program relating Gromov–Witten invariants and birational geometry. The key step in this program is a hypothetical formula expressing genus 0 GW invariants of a blow-up of a smooth variety X at smooth center Y, in terms of those of X and Y. The exact shape of the formula turns out to be quite complicated, involving WKB solutions of certain universal family of linear ODEs depending only on one natural number, the codimension of Y in X. In his talk, Kontsevich will give the exact formulation of the blow-up formula (which is joint work in progress with Ludmil Katazrkov and Tony Yu Yue). Hypothetically, this formula extends to the equivalence between Fukaya categories of the blow-up and of the disjoint union of X and several copies of Y.

Tsung-Ju Lee
Harvard University

Mirror Symmetry on Singular Calabi–Yau Spaces

Mirror symmetry for singular Calabi–Yau spaces were discovered by Hosono, Lian, Takagi and Yau in their recent work on K3 surfaces. In this talk, Lee will construct pairs of singular Calabi–Yau varieties arising from double covers and give some numerical evidence that indicates they are mirror pairs, including Euler characteristics, Hodge numbers and orbifold Gromov–Witten invariants. Lee will also discuss these pairs from categorical perspectives.

Daniel Pomerleano
University of Massachusetts, Boston

The Frobenius Property and HMS for log CY Varieties

Pomerleano will discuss a symplectic version of the Frobenius structure conjecture, recently proven by Keel and Yu. He will then explain how this result, together with recent finiteness results for wrapped invariants of affine log Calabi–Yau varieties, allows one to prove HMS for a wide class of examples.

Paul Seidel
Massachusetts Institute of Technology

Mirror Symmetry and Noncommutative Linear Systems

Seidel will try to survey the picture of HMS relative to an anticanonical divisor, and what this implies for the Fukaya categories of Calabi–Yau hypersurfaces.

• Simons Foundation Lecture: November 17, 2021

5:00-6:15 PM

A public lecture on meeting themes will be presented the evening before the official start of the conference. Participants are encouraged to register and attend should their travel and other schedules align.

Simons Foundation Lectures are free public colloquia related to basic science and mathematics. These high-level talks are intended for professors, students, postdocs and business professionals, but interested people from the metropolitan area are welcome as well.

Participation is optional; separate registration is required.

Mina Aganagic
University of California, Berkeley

(Homological) Knot Invariants From Mirror Symmetry

Quantum invariants of knots (also known as Chern-Simons knot invariants) have many applications in mathematics and physics. For example, Khovanov showed in 1999 that the simplest such invariant, the Jones polynomial, arises as the Euler characteristic of a homology theory. The knot categorification problem is to find a general construction of knot homology groups and to explain their meaning: What are they homologies of?

Mirror symmetry is another important strand in the interaction between mathematics and physics. Homological mirror symmetry, formulated by Kontsevich in 1994, naturally produces hosts of homological invariants. Sometimes, it can be made manifest, and then its striking mathematical power comes to the fore. Typically, though, it leads to invariants that have no particular interest outside of the problem at hand.

In this lecture, Mina Aganagic will present how she recently showed there is a vast new family of mirror pairs of manifolds, for which homological mirror symmetry can be made manifest. They do lead to interesting invariants. In particular, they solve the knot categorification problem.

To attend this in-person event, you will need to register in advance and provide:

Acceptable proof of vaccination (vaccine card/certificate, a copy or photo of vaccine card/certificate or electronic NYS Excelsior Pass or NJ Docket Pass)
Photo ID
Eventbrite ticket confirmation email with QR code
Simons Foundation Health Screening Questionnaire approval email

Entrance will not be granted without this documentation.

On-site registration will not be permitted. Walk-in entry will be denied.

• Participation & Funding

Participation in the meeting falls into the following four categories. An individual’s participation category is communicated via their letter of invitation.

Group A – PIs and Speakers
The foundation will arrange and pay for all air and train travel to the conference as well as hotel accommodations and reimbursement of local expenses. Business-class or premium economy airfare will be booked for all flights over five hours.

Group B – Out-of-town Participants
The foundation will arrange and pay for all air and train travel to the conference as well as hotel accommodations and reimbursement of local expenses. Economy-class airfare will be booked for all flights.

Group C – Local Participants
Individuals in Group C will not receive financial support, but are encouraged to enjoy all conference-hosted meals.

Group D – Remote Participants
Individuals in Group D will participate in the meeting remotely. Please register at the link above and a remote participation link will be sent to you approximately two weeks prior to the meeting.

• Travel & Hotel

Air and Train
The foundation will arrange and pay for all air and train travel to the conference for those in Groups A and B. Please provide your travel specifications by clicking the registration link above. If you are unsure of your group, please refer to your invitation sent via email.

Personal Car
For participants in Groups A & B driving to Manhattan, The James NoMad Hotel offers valet parking. Please note there are no in-and-out privileges when using the hotel’s garage, therefore it is encouraged that participants walk or take public transportation to the Simons Foundation.

Hotel
Participants in Groups A & B who require accommodations are hosted by the foundation for a maximum of three nights at The James NoMad Hotel. Any additional nights are at the attendee’s own expense. To arrange accommodations, please register at the link above.

22 E 29th St
New York, NY 10016
(between 28th and 29th Streets)

• COVID-19 Policy

ALL in-person meeting attendees must be vaccinated against the COVID-19 virus with a World Health Organization approved vaccine, be beyond the 14-day inoculation period of their final dose, and provide proof of vaccination upon arrival to the conference. Acceptable vaccines can be found at the bottom of this page on WHO’s site.

• Reimbursement

Individuals in Groups A & B will be reimbursed for meals not hosted by the Simons Foundation as well as local expenses, including ground transportation. Additional information in this regard will be emailed on the final day of the meeting.

• Contacts

Travel & Hotel Assistance
Ovation Travel Team
sfnevents@ovationtravel.com
www.ovationtravel.com

Meeting Questions & Assistance
Meghan Fazzi
Manager, Events and Administration, MPS, Simons Foundation
mfazzi@simonsfoundation.org
(212) 524-6080

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