### Homological Mirror Symmetry and Topological Recursion, Miami, January 27-February 1, 2020

Speakers:

Maxim Kontsevich

Denis Auroux

Bertrand Eynard

Mohammed Abouzaid

Jørgen Ellegaard Andersen

Matthew Ballard

Roman Bezrukavnikov

Ron Donagi

Yu-Wei Fan

Alexander Goncharov

Sergey Gukov

Yuan-Pin Lee

Ernesto Lupercio

Emmy Murphy

Tony Pantev

Alexander Polishchuk

Artan Sheshmani

Bernd Siebert

Yan Soibelman

Abigail Ward

**Organizers:** D. Auroux, L. Katzarkov, M. Kontsevich, E. Lupercio, T. Pantev

This event is partially supported by the Simons Collaboration on Homological Mirror Symmetry, with the assistance of the University of Miami Department of Mathematics. The programme will feature 3 mini-courses and a range of research talks in various areas of homological mirror symmetry and related topics.

This year’s mini-courses will be given by Maxim Kontsevich, Bertrand Eynard and Denis Auroux.

**Venue:** *IMPORTANT: Due to capacity constraints, the venue may be moved. Updated information will be given once this is determined.* The conference is scheduled to take place at the Marriott Courtyard Coral Gables, from Monday morning until Saturday lunchtime.

**Airport:** Miami International Airport is about 7 miles away. The most convenient way to reach the conference venue is to take a taxi. Participants expecting reimbursement: please keep all your original boarding passes since they may be needed for reimbursement.

**Registration:** There will be a registration fee of $40 to cover the cost of refreshments at the conference. A registration web form has been set up in order to allow participants to register, pay the registration fee, and request a hotel reservation. Please register here.

### Tentative schedule

Monday January 27

9:30-10:30 | M. Kontsevich: TBA 1 |

11:00-12:00 | Y. Soibelman: TBA |

2:00-3:00 | D. Auroux: Mirrors of curves and their Fukaya categories I |

3:30-4:30 | M. Abouzaid: TBA |

5:00-6:00 | R. Donagi: TBA |

Tuesday January 28

9:30-10:30 | D. Auroux: Mirrors of curves and their Fukaya categories II |

11:00-12:00 | B. Siebert: TBA |

2:00-3:00 | M. Kontsevich: TBA 2 |

3:30-4:30 | A. Polishchuk: TBA |

5:00-6:00 | R. Bezrukavnikov: TBA |

Conference dinner? |

Wednesday January 29

9:30-10:30 | D. Auroux: Mirrors of curves and their Fukaya categories III |

11:00-12:00 | E. Murphy: TBA |

2:00-3:00 | A. Goncharov: TBA 1 |

3:30-4:30 | A. Ward: TBA |

5:00-6:00 | A. Sheshmani: Higher rank flag sheaves on projective surfaces, and relations to Vafa-Witten and Kapustin-Witten invariants for complex 3 and 4 folds |

Thursday January 30

9:30-10:30 | B. Eynard: Topological recursion: a B-model method to compute Gromov-Witten invariants and enumerative geometry I |

11:00-12:00 | Y.-P. Lee: Functoriality in Gromov-Witten theory |

2:00-3:00 | M. Kontsevich: TBA 3 |

3:30-4:30 | M. Ballard: Exceptional collections and rationality |

Friday January 31

9:30-10:30 | T. Pantev: The pushforward theorem and its applications |

11:00-12:00 | J.E. Andersen: TBA 1 |

2:00-3:00 | B. Eynard: Topological recursion: a B-model method to compute Gromov-Witten invariants and enumerative geometry II |

3:30-4:30 | S. Gukov: A proposal for open DT invariants |

5:00-6:00 | E. Lupercio: Remarks on the moduli space in quantum toric geometry |

Saturday February 1

9:00-10:00 | B. Eynard: Topological recursion: a B-model method to compute Gromov-Witten invariants and enumerative geometry III |

10:30-11:30 | Y.-W. Fan: New rational cubic fourfolds via Cremona transformation |

12:00-1:00 | J.E. Andersen: TBA 2 |

### Titles and Abstracts

D. Auroux: Mirrors of curves and their Fukaya categories

**Abstract:** These talks will focus on homological mirror symmetry for curves, and in particular, the symplectic geometry of their mirrors (i.e., comparing coherent sheaves on a curve to a suitable Fukaya category of the mirror). In the first talk, we will review the construction of Landau-Ginzburg models mirror to curves in (**C***)^{2} or in toric surfaces, and the notion of fiberwise wrapped Fukaya category, as well as computations and verifications of HMS in this framework (this is joint work with Mohammed Abouzaid). We will also discuss the geometric relationship between the smooth fiber of the Landau-Ginzburg model, its singular fiber, and its total space, and the corresponding functors between their Fukaya categories (touching on work in progress by Maxim Jeffs). These results also apply to hypersurfaces in higher-dimensional toric varieties, in abelian varieties (Catherine Cannizzo’s thesis), and to complete intersections.

We will then discuss more speculative ways of viewing the symplectic geometry of the mirror from a lower-dimensional perspective: first in terms of “tropical Lagrangians” in the fiber of the Landau-Ginzburg model (using ideas of Andrew Hanlon and Jeff Hicks), and then in terms of geometry inside the critical locus (work in progress with Alexander Efimov and Ludmil Katzarkov). For mirrors of curves, this leads to a new flavor of Lagrangian Floer theory in trivalent configurations of Riemann surfaces, which we will describe in detail as well as the manner in which it relates to the geometry of the curve.

M. Ballard: Exceptional collections and rationality

**Abstract:** The general expectation, attributed to Orlov, is that a smooth projective variety with a full exceptional collection must be rational over its base field. We will discuss this question over non-closed fields. We will show that there exists smooth projective geometrically rational 3-folds which possess full etale-exceptional collections (where End(E) is a finite separable extension of the base field) but not any points over the base field. In the other direction, we will show the expectation holds for smooth projective toric varieties over any base field: if a smooth projective (neutral) toric variety over k possesses a full k-exceptional collection then it is in fact k-rational. This is joint work with A. Duncan, A. Lamarche, and P. McFaddin.

B. Eynard: Topological recursion: a B-model method to compute Gromov-Witten invariants and enumerative geometry

Y.-W. Fan: New rational cubic fourfolds via Cremona transformation

**Abstract:** It is conjectured that two cubic fourfolds are birational if their associated K3 categories are equivalent. We prove this conjecture in the case when one of the cubic fourfolds contains a Veronese surface. The main technique is Cremona transformation on P^{5}. Using the same technique, we find new rational cubic fourfolds. Joint work with Kuan-Wen Lai.

S. Gukov: A proposal for open DT invariants

Y.-P. Lee: Functoriality in Gromov-Witten theory

**Abstract:** I will discuss the functoriality problem in Gromov–Witten theory, and explain a series of results in this context, including quantum Lefschetz, quantum Leray-Hirsch, as well as the functoriality with respect to birational transformations.

E. Lupercio: Remarks on the moduli space in quantum toric geometry

**Abstract:** In this talk I will define the main objects appearing in quantum toric geometry (which quantize classical toric geometry): Quantum toric stacks and their moduli spaces. Then, I will explain how to put a natural “complex structure of to homotopy” on the moduli space.

T. Pantev: The pushforward theorem and its applications

**Abstract:** I will discuss the notion of a relative shifted symplectic structure along the stalks of a constructible sheaf of derived stacks on a stratified space. I will describe a general pushforward theorem producing relative shifted symplectic forms and will explain explicit techniques for computing such forms. As an application I will describe a universal construction of Poisson structures on derived moduli of Stokes data on smooth varieties and will explain how symplectic leaves arise from fixing irregular types and local formal monodromies at infinity. This is a joint work with Dima Arinkin and Bertrand Toën.

A. Sheshmani: Higher rank flag sheaves on projective surfaces, and relations to Vafa-Witten and Kapustin-Witten invariants for complex 3 and 4 folds

**Abstract:** We study moduli space of holomorphic triples $f: E_{1} → E_{2}$, composed of (possibly rank $>1$) torsion-free sheaves $(E_{1}, E_{2})$ and a holomorphic map between them, over a smooth complex projective surface $S$. The triples are equipped with a Schmitt stability condition. We prove that when the Schmitt stability parameter becomes sufficiently large, the moduli space of triples benefits from having a perfect relative and absolute obstruction theory in some cases (depending on Chern character of $E_{1}$). We further generalize our construction to higher-length flags of higher rank sheaves by gluing triple moduli spaces, and extend earlier work, with Gholampur and Yau, where the obstruction theory of nested Hilbert schemes over the surface was studied. Here we extend the earlier results to the moduli space of flags $E_{1}→ E_{2}→ … → E_{n}$, where the maps are injective (by stability). There is a connection, by wall-crossing in the master space, developed by Mochizuki, between the theory of such higher rank flags, and the theory of Higgs pairs on the surface, which provides the means to relate the flag invariants to the local DT invariants of a threefold given by a line bundle over the surface, $X :=Tot(L → S)$. The latter DT invariants, when L is the canonical bundle of S, contribute to Vafa-Witten invariants. Joint work with Shing-Tung Yau, arXiv:1911.00124. If the time permits, I will also talk about some more generalizations of this construction to local surface 4-folds and connections to Kapustin-Witten theory.