# FRG workshop on Brauer groups and derived categories

## 20-22 October 2023 @ Northwestern

# What

A highly focused workshop on recent developments in Brauer groups and derived categories, suitable for people working in the field, organized around a mini-course by each speaker.

# Speakers

James Hotchkiss (Columbia)

Alex Perry (Michigan)

Laura Pertusi (Milano)

# Schedule

F | Sa | Su |
---|---|---|

1500 Pertusi 1 | 0915 Pertusi 2 | 0900 Hotchkiss 3 |

1545 Coffee | 1030 Coffee | 1000 Coffee |

1615 Perry 1 | 1115 Hotchkiss 2 | 1030 Pertusi 3 |

1700 Hotchkiss 1 | 1230 Lunch | 1130 Coffee |

1745 Day ends | 1430 Perry 2 | 1200 Perry 3 |

1545 Day ends | 1300 Workshop ends |

# Series titles and abstracts

**Hotchkiss and Perry**, *The period-index conjecture*.
The period-index conjecture asks for a precise bound on one measure of complexity of a Brauer class on a variety (its index) in terms of another (its period). The goal of this mini-course is to explain some recent progress on this problem, based on a mixture of ideas from Hodge theory, noncommutative/derived algebraic geometry, and enumerative geometry. In particular, we will show how to interpret the problem as an instance of the integral Hodge conjecture for categories, which in turn can be studied using Donaldson-Thomas counting invariants, leading to a proof of the period-index conjecture for abelian threefolds.

**Pertusi**, *Bridgeland stability conditions and families of hyperkaehler manifolds*.
A hyperkaehler manifold is a compact complex simply connected Kaehler manifold whose space of holomorphic two-forms is generated by a symplectic form, unique up to scalar multiplication. Together with complex tori and irreducible Calabi–Yau manifolds, they are building blocks for compact Kaehler manifolds with trivial first Chern class.

In dimension two, hyperkaehler manifolds are K3 surfaces, while finding examples in higher dimensions is a challenging problem.

The aim of this mini-course is to investigate the recent developments about the construction of families of hyperkaehler manifolds, using derived categories techniques and moduli spaces of semistable objects.

# Slides

Hotchkiss 1, Hotchkiss 2, Hotchkiss 3, Perry 1, Perry 2, Perry 3, Pertusi 1, Pertusi 2, Pertusi 3.

# Organizers

Benjamin Antieau, Andrei Caldararu, Max Lieblich, Akhil Mathew, Martin Olsson

# Registration

Register here no later than 2 October 2023; for funding, register no later than 14 September 2023.

# Funding

Some funding is available for graduate students, early-career researchers, and others. To apply for it, your
registration must be received by 14 September 2023, with decisions made by 21 September 2023.
For funding, graduate students must have a letter of recommendation submitted on their behalf to me,
at `antieau@northwestern.edu`

, no later than the deadline above.

# Lodging

Participants, even those we fund, are asked to book their own lodging and be reimbursed. We will book lodging for speakers. Suitable hotels include the Hyatt House Evanston and the Hilton Orrington Evanston.

*Added 31 August*: the weekend of the workshop is Family Weekend at Northwestern. As a consequence,
hotel rooms are scarce and expensive. If you have not already booked, you should do so soon. If
that fails, you might look on airbnb or book a hotel in downtown Chicago, near Oglivie
Station. From there, you can take the Metra UP-N Line
easily to Evanston. The train runs every 30 minuts at rush hour and every 60 minutes at
off-peak times, but takes only 23 minutes to get to the Evanston Davis stop, from which campus
is a 15 minute walk.

# Travel

Northwestern is accessible via Amtrak through Chicago Union Station and via plane through Chicago O’Hare or
Midway airports. One can take a cab from any of these stations or take public transportation on the Red or
Purple ‘L’ Lines or on the **UP-N
Line** of the Metra.

# Location

Annenberg Hall G21 on Friday afternoon. Swift Hall 107 on Saturday and Sunday.

# Restaurants

There are no official meals planned for this workshop. Recommended Evanston restaurants for dinner include Peppercorns, Prairie Moon, LeTour (reservations recommended), Union Squared Pizza (reservations recommended), Tapas Barcelona, and Ridgeville Tavern. There are many others I have not tried.

# Parking

Parking is available at either the South or North Campus Parking Garage for $9/day. The South Garage is a 5-10 minute walk from the location of the conference, the North Garage is a 15-20 minute walk. For more details, see here.

# Acknowledgments

This conference is funded by the NSF grant DMS-2152235, *FRG: higher categorical structures in
algebraic geometry*.
The image above was created at
deepai.org with the prompt
“Brauer group”.