# FRG Workshop on Stability, Moduli Spaces and Applications UIC December 6-8, 2019

Speakers:

Matteo Altavilla (University of Utah)
Katrina Honigs (University of Oregon)
Eric Larson (Stanford University)
Sayanta Mandal (University of Illinois at Chicago)
Alina Marian (Northeastern University)
Alex Perry (Columbia University)
Marin Petkovic (University of Utah)
Ryan Takahashi (University of Oregon)
Isabel Vogt (Stanford University)

Organizers: İzzet Coşkun (U. Illinois at Chicago), Howard Nuer (U. Illinois at Chicago), Aaron Bertram (U. Utah), Jack Huizenga (Penn State U.), Emanuele Macri (Northeastern U.)

Registration: Registration is free and open to everyone. If you are planning to attend, please register by October 30, 2019 by filling out the registration form here. There may be limited funds available for graduate students and post-docs. If you would like to be considered for financial aid, please apply by October 15, 2019 and have a short recommendation letter from your advisor be forwarded to uicagworkshops@gmail.com

Registered participants: You can find a list of registered participants here

Schedule:
Friday:
4:10-5:10 Katrina Honigs
5:15-6:00 Marin Petkovic

Saturday:
9:00-10:00 Alina Marian
10:15-11:15 Alina Marian
11:30-12:15 Matteo Altavilla
2:00-2:45 Ryan Takahashi
2:55-3:40 Sayanta Mandal
4:00-5:15 Alex Perry

Sunday:
9:30-10:30 Isabel Vogt
10:30-11:30 Eric Larson

The lectures will take place at Lecture Center C3. See the campus map for directions
There will be light refreshments at 8:30 on Saturday and 9:00 on Sunday at SEO 300.

Titles and Abstracts:

Matteo Altavilla:
A Kuznetsov-Torelli theorem for the quartic double solid
A smooth Fano 3fold Y of Picard rank 1 and index 2 admits a non-trivial Kuznetsov component K(Y) in D(Y). One can ask whether the existence of an equivalence of categories between Ku(Y) and Ku(Y') implies that Y and Y' are isomorphic. I will present a proof of this result in the case d=2, which arises as an application of a general moduli construction for Y's in all degrees. This is joint work with M. Petkovic and F. Rota.

Katrina Honigs:
Rational points and derived equivalence
In this talk, I will discuss recent joint work with Addington, Antieau and Frei where we gave the first examples of derived equivalences between smooth, projective varieties where one variety has a Q-rational point and the other does not, including a particular example of derived equivalent hyperkahler 4-folds.

Eric Larson:
The integral Chow ring of $\bar{M}_2$
In this talk, we will compute the Chow ring of the moduli stack of stable curves of genus 2 with integral coefficients.

Sayanta Mandal:
Betti numbers of the moduli space of stable sheaves on the projective plane
We will discuss stabilization of Betti numbers of moduli space of stable sheaves on surfaces, and in the special case of the projective plane we will produce lower bounds on the second Chern class of the sheaves such that the Betti numbers of their moduli space stabilize.

Alina Marian:
On the Chow ring of holomorphic symplectic manifolds
I will propose a series of basic conjectural identities in the Chow rings of holomorphic symplectic manifolds of K3 type, and will discuss evidence for them. The emerging structure naturally generalizes in higher dimensions a set of key properties of cycles on a K3 surface. I will initially provide background and context for this circle of ideas. The talks are based on joint work with Ignacio Barros, Laure Flapan, and Rob Silversmith.

Alex Perry:
Homological projective geometry
I will discuss some surprising "homological" counterparts of constructions and results in classical projective geometry. This gives a powerful framework for producing varieties whose derived categories are equivalent, or more generally have a large subcategory in common. This is joint work with Alexander Kuznetsov.

Marin Petkovic
Moduli spaces on Kuznetsov component of Fano threefolds of index 2
We study induced Bridgeland stability conditions on the Kuznetsov components of the derived categories of Fano threefolds of Picard rank one and index two. We construct projective moduli spaces of stable objects that contain the threefold as subvarieties. This is joint work with M. Altavilla and F. Rota.

Ryan Takahashi:
A categorical sl_2 action on some moduli spaces of sheaves
Certain spherical twists on K3 surfaces induce stratified Mukai flops between moduli spaces of stable sheaves, modeled locally by correspondences between cotangent bundles to Grassmannians. Cautis, Kamnitzer, and Licata have shown how to construct a geometric categorical sl_2 action in this local model. I will explain that such an action can also be obtained globally on the moduli spaces. In particular, this implies that the birational moduli spaces in this setup are also derived equivalent.

Isabel Vogt:
Stability of normal bundles of space curves
In this talk we will introduce the basic notion of stability of vector bundles on curves and how it can be approached by degenerating the curve. We'll then apply this to the bundle that controls the deformation theory of a curve embedded in projective space: the normal bundle. This is joint work with Izzet Coskun and Eric Larson.

Travel Information:

The conference will be held in the UIC Math Department, which is located in the Science and Engineering offices building (SEO, 851 S Morgan St, Chicago, IL 60607). There will be signs with directions to the lecture room.

Traveling by car: UIC is easily accessible by car, and is located just off of either the 90/94 or 290 interstates. If you are coming into UIC, it's best to park at the Halsted Parking Structure at the NE corner of the Taylor and Halsted intersection.

Traveling by train/bus: UIC is also close to Chicago's Union Station with convenient Amtrak trains / Megabus routes to many locations in the region. A cab to UIC from Union Station should cost under $10. The Greyhound station is also very close by, and an easy walk to UIC. Traveling by air: Chicago has two major airports, O'Hare (ORD) and Midway (MDW), and it is easy to get to UIC from either one. If you are flying into O'Hare, you can take a Blue Line train (costing$5) into the city (towards the loop or Forest Park) which stops right next to the UIC campus (at the UIC/Halsted station). For reference, a taxi from O'Hare to downtown will cost around $45 before tip (and can be paid with a credit card). If you are flying into Midway, you can take an Orange Line train (costing$2.50) into the city (towards the loop), which will allow you to transfer to the Blue Line (at the Harold Washington Library station) which stops right next to the UIC campus (at the UIC/Halsted station). For reference, a taxi from Midway to downtown will cost around \$25 before tip (and can be paid with credit card).

Accommodations: We will make hotel reservations directly for speakers and participants receiving support from the conference. All other participants should make their own arrangements. We will not call you and ask for your credit card information. Please beware of scams.

Important Information about Travel Reimbursement : Visitors who will receive funding from NSF must fly a US carrier in order to receive airfare reimbursement. When US carriers have code-share agreements with foreign carriers, you may use a foreign carrier only if the flight number has a US carrier name. If you are not a US citizen or permanent resident, please do not make your own travel arrangements, but contact us for travel arrangements.

We are greatful for the generous support of the University of Illinois at Chicago, the Department of Mathematics, Statistics, and Computer Science, the Visitors' Fund and the National Science Foundation. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.