About

The workshop will begin on the afternoon of Friday, April 29 and go through Sunday, May 1.

Keep up-to-date on Cornell Covid information and information for visitors by going to Cornell Covid Information and Cornell Covid Visitor Information.

Workshop speakers

Lev Borisov, Rutgers University

Title: Equivariant derived categories of smooth toric DM stacks (Video)

Abstract: Derived category of equivariant coherent sheaves on a toric DM stack is in many ways simpler than its non-equivariant counterpart, but it also captures less information. I will talk about a joint paper with D. Orlov devoted to study of these categories.

Andrei Căldăraru, University of Wisconsin, Madison

Title: Categorical Enumerative Invariants and the \lambda_g conjecture (Notes)

Abstract: I will give an overview of the theory of Categorical Enumerative Invariants (CEIs), as developed in joint work with Junwu Tu and Kevin Costello. The emphasis will be on the role played by the Givental group, which controls the effect of changes of splitting on the CEIs.

If time allows I will state two new conjectures which relate the derived category of the nodal cubic with the classical $\lambda_g$ conjecture in Gromov-Witten theory, due to Getzler, Faber and Pandharipande. The new work is joint with Weng-Him Cheung and Yunfan He.

Katrina Honigs, Simon Fraser University

Title: Symplectic involutions of hyperk\"ahler4-folds of Kummer type and derived equivalences (Video)

Abstract: Examples of hyperk\"ahler varieties of Kummer type may be constructed as fibers of the Albanese map on moduli of stable sheaves on an abelian surface with fixed Mukai vector. This construction generalizes that of Kummer K3 surfaces to higher dimensions. Hassett and Tschinkel showed that a portion of the middle cohomology of generalized Kummer 4-folds -- those where the moduli space coincides with the Hilbert scheme of length 3 points -- may be understood as fixed loci of symplectic involutions. In recent work with Sarah Frei, we have extended this result by finding explicit descriptions of symplectic involutions on these 4-folds of Kummer type, allowing for a characterization of the Galois action on the cohomology when working over non-closed fields. I will discuss relationships between this work and derived equivalences. If there is time, I will also discuss the geometry of a particular fixed locus in an example.

Kimoi Kemboi, Cornell University

Title: Full exceptional collections on rank-two linear GIT quotients (Video)

Abstract: We will discuss the existence of full strong exceptional collections consisting of vector bundles on GIT quotients of certain linear actions of a split reductive group G of rank two. These vector bundles come from irreducible G-representations whose weights lie in a “window” in the weight space of G. This is based on joint work with Daniel Halpern-Leistner.

Alicia Lamarche, The University of Utah

Title: Derived Categories and Rational Points for a class of toric Fano varieties (Video)

Abstract: I will discuss forthcoming work with Matthew Ballard on using the derived category of coherent sheaves to detect the existence of rational points on a particular family of arithmetic toric Fano varieties. More precisely, I will explain how we show that a member of this family of varieties is rational if and only if its bounded derived category of coherent sheaves admits a full étale exceptional collection.

Alina Marian, Northeastern University

Title: On the cohomology of Quot schemes parametrizing torsion quotients on a curve. (Video)

Abstract: I will explain features of the cohomology groups of curve Quot schemes, discussing in particular the Quot scheme of rank zero quotients of a fixed locally free sheaf on a smooth projective curve. This is a simpler setting for phenomena occurring also in the geometry of moduli spaces of stable sheaves on surfaces.

Davesh Maulik, MIT

Title: Higgs bundles and the P=W conjecture (Video)

Abstract: Given a compact Riemann surface C, nonabelian Hodge theory relates topological and algebro-geometric objects associated to C. Specifically, complex representations of the fundamental group \pi_1(C) are in correspondence with algebraic vector bundles on C, equipped with an extra structure called a Higgs field. This gives a transcendental matching between two very different moduli spaces for C: the character variety of \pi_1(C) (parametrizing its representations) and the so-called Hitchin moduli space of C (parametrizing vector bundles with Higgs field).

In 2010, de Cataldo, Hausel, and Migliorini proposed a conjectural relation -- now called the P=W conjecture -- between these two spaces. This conjecture gives a precise link between the topology of the Hitchin space and the Hodge theory of the character variety, imposing surprising constraints on each side.

In the first part of this talk, I will give an introduction to this circle of ideas; in the second part, I will survey some recent progress towards understanding this conjecture, using ideas from compact hyperkahler geometry and geometry in characteristic p.

Martin Olsson, University of California, Berkeley

Title: Relativizing derived equivalences (Video)

Abstract: Derived equivalences between smooth projective varieties have to respect various structures, such as Serre functors and actions of certain line bundles. In this talk I will discuss some foundational results that enable one in such situations to conclude that the kernel defining the derived equivalence is given by a family of kernels relative to a positive dimensional base. Joint work with Max Lieblich.

Alexander Polishchuk, University of Oregon

Title: Poisson structures on moduli spaces associated with an elliptic curve and Cremona transformations (Video)

Abstract: I will talk about the modular point of view on Feigin-Odesskii Poisson brackets on projective spaces. This point of view leads to some interesting birational transformation between projective spaces. I will show how these Poisson brackets can be computed in terms of Massey products. This leads to explicit formulas for the corresponding Cremona transformations.

Workshop Schedule

Breakfast and coffee breaks will be held in 532 Malott Hall

Friday, April 29th: [Talks in 228 Malott Hall, Bache Auditorium]

12:00-1:30 Registration Check-in 532 Malott Hall

1:40-2:25 Learning seminar: Bondal-Orlov's Reconstruction Theorem (Rachel Webb)

2:40-3:25 Learning seminar: The Fourier-Mukai transform (Tudor Pădurariu)

3:30-4:00 Department tea and Registration check-in in 532 Malott Hall

4:00-5:40 Davesh Maulik - Higgs bundles and the P=W conjecture

(first half joint with department colloquium)

6:00-8:00 Conference dinner

Saturday, April 30th: [Talks in 251 Malott Hall]

8:30-9:00 Breakfast

9:00-10:00 Alexander Polishchuk - Poisson structures on moduli spaces associated with an elliptic curve and Cremona transformations

10:10-11:10 Katrina Honigs - Symplectic involutions of hyperkähler4-folds of Kummer type and derived equivalences

Coffee break

11:40-12:40 Kimoi Kemboi - Full exceptional collections on rank-two linear GIT quotients

Lunch break

1:40-2:40 Lev Borisov - Equivariant derived categories of smooth toric DM stacks

4:00-6:00 Picnic dinner at the Stewart Park Large Pavilion.

Sunday, May 1st: [Talks in 251 Malott Hall]

8:30-9:00 Breakfast

9:00-10:00 Martin Olsson - Relativizing derived equivalences

10:10-11:10 Alicia Lamarche - Derived Categories and Rational Points for a class of toric Fano varieties

Coffee break

11:40-12:40 Andrei Caldararu - Categorical Enumerative Invariants and the \lambda_g conjecture

Lunch break

1:40-2:40 Alina Marian - On the cohomology of Quot schemes parametrizing torsion quotients on a curve.

Learning Seminar Talks

Rachel Webb, University of California, Berkeley

Title: Bondal-Orlov's Reconstruction Theorem
Abstract: Let X be a smooth irreducible projective variety whose dualizing sheaf is either ample or anti-ample, and let Y be any smooth projective variety. A theorem of Bondal and Orlov states that if the bounded derived categories of X and Y are equivalent as graded categories, then X and Y are isomorphic. We will discuss their argument, paying close attention to the role of Serre duality in the proof and its reincarnation as a Serre functor.

Tudor Pădurariu, Columbia University

Title: The Fourier-Mukai transform
Abstract: Fourier-Mukai transforms are natural functors between derived categories of coherent sheaves on smooth projective varieties. By a theorem or Orlov, every derived equivalence between such categories is induced by a Fourier-Mukai transform. The plan for the talk is to define the Fourier-Mukai transform, discuss its properties, and give examples of equivalences between derived categories of coherent sheaves on smooth projective varieties, following Mukai, Bridgeland-King-Reid, and Bridgeland.

Registration

Please go to the following link and complete the questionnaire to register: https://cornell.ca1.qualtrics.com/jfe/form/SV_0fDn54sP5fMqU8S

We have limited funds to support participants. Please indicate in the registration form if you would like to be considered for funding. Early career mathematicians and people from underrepresented groups are particularly encouraged to apply.

The registration deadline is March 4, 2022 for those applying for funding. Decisions on financial support will be made by March 25th.

The general registration deadline is April 15, 2022.

Housing Accommodations

Room blocks have been reserved at the following hotels. All participants except for invited speakers are asked to make their own reservations. Early reservations are encouraged, due to high seasonal demand. Unreserved rooms in the contracted room blocks including the Cornell rates will be released for general sale on the dates specified. If possible, we ask that you make your reservations as soon as you possibly can. Thank you.

1. Hilton Garden Inn Ithaca
   Rate: $199.00/night
   To make your reservation: Guests can call 607-277-8900 or 877-STAY-HGI (toll-free) and ask for the group block, OR go online to www.ithaca.hgi.com and enter Group/Convention code: MATHWS
   Reservations must be made by April 8, 2022

2. Homewood Suites by Hilton Ithaca
   Rate: $159.00/night
   To make your reservation go to https://www.my-event.hilton.com/ithhwhw-frg-d34f0108-f50c-4239-9ebc-f601fa24eb23/
   Reservations must be made by March 29, 2022
   Reservation link will be posted soon.
   

Covid-19 Policies

We will adhere to Cornell's Covid-19 policies at the time of the event (Covid Guidelines for visitors and Covid Guidelines for events). Currently, Cornell requires visitors to present a proof of either vaccination or a recent negative Covid test.

Inclusivity

Cornell University is committed to providing a safe, inclusive, and respectful learning, living, and working environment. To this end, Cornell will not tolerate sexual and related misconduct. Through Cornell University Policy 6.4, and the applicable procedures, the university provides means to address bias, discrimination, harassment, and sexual and related misconduct, including gender-based harassment, sexual harassment, sexual assault, domestic and dating violence, stalking, and sexual exploitation. Reports of bias, discrimination, and harassment can be made online at www.biasconcerns.cornell.edu or by contacting the Office of the University Title IX Coordinator at titleix@cornell.edu.

Organizers

• Daniel Halpern-Leistner

• Johan de Jong

• Alex Perry

• Giulia Saccà

• Xiaolei Zhao

Please send any questions to daniel.hl@cornell.edu.