Workshop on Algebra and Representation Theory, Held on Oregonian Grounds

7 August - 11 August 2017

University of Oregon

Eugene, OR

In the 70's, Bernstein, Gelfand and Gelfand defined a category of modules over a simple Lie algebra with many remarkable properties, called category **O**. Around 1990, Soergel made the remarkable observation that this category has a surprising symmetry: the endomorphism algebra of the sum of all indecomposable projective modules is the same as the Ext-algebra of the sum of all simple modules. In fancier language, category **O** is Koszul self-dual. His proof is both beautiful and dissatisfying: using certain bimodules named in his honor, he showed that an intermediate, more combinatorial algebra is isomorphic to both the endomorphisms and the Exts.

This observation about category **O** seems to be a special case of a more general picture which sometimes goes by the name *symplectic duality*. There are category **O**'s attached not just to Lie algebras, but to conical symplectic singularities. In this workshop, we'll describe a second situation where we find a similar Koszul duality, between the category **O**'s associated to two different varieties: the Higgs and Coulomb branch of an N=4 supersymmetric gauge theory. In this workshop, we'll concentrate on the abelian case, which was worked out by Braden, Proudfoot, Licata and Webster. It is considerably simpler, but preserves many of the most important features of the general case.

We will focus our attention on the following papers:

Gale duality and Koszul Duality by Braden, Licata, Proudfoot, Webster.

Hypertoric category O by Braden, Licata, Proudfoot, Webster.

Towards a mathematical definition of Coulomb branches of 3-dimensional N=4 gauge theories II by Braverman, Finkelberg, Nakajima.

Koszul duality between Higgs and Coulomb categories O by Webster.

The workshop will be led by Ben Webster, and will consist of a combination of lectures and problem sessions.

It will be organized by Ben Elias, under the watchful eye of Nicholas Proudfoot.

If you are interested in participating, email Ben Elias.
Please include your school, advisor, and a brief description of your research interests. Please place [WARTHOG] in the subject line of your email.

Funding for accommodations in Eugene (but not for travel) will be available to students and postdocs as long as space and funds remain. Let Ben know if you have any special needs or requests.

If you are on this list then I am expecting you to come to the workshop. Unless you have an asterisk by your name, accommodations will be provided to you.

In construction.

General Schedule: There will be lectures and exercises Monday through Friday, from approximately 9AM to 5PM, with gaps between and a space for lunch. Wednesday is a half-day, with the afternoon being left open for an organized hiking trip to nearby Spencer Butte. Wednesday evening we will have a pizza party at a local park, sponsored by the workshop. On Monday, Tuesday, and Thursday evening there will be an optional, organized exercise session.

Food: Starting at 830AM each day of the workshop, coffee and pastries will be catered to the lecture room. Aside from this and the pizza party, participants will have to pay for their own food. Carson residence hall has a cafeteria with more substantial breakfast for purchase. There will be time to eat lunch at many local restaurants, and restaurant recommendations will be posted at a later date.

Accommodations: Participants will be staying in the Earl residence hall on campus. The cost of this will be covered by the workshop.

Accommodations and arrival/departure timing: It is assumed that participants will arrive in Eugene on Sunday before the workshop, and leave on either Friday night or Saturday after the workshop. Housing will be available for participants leaving on Sunday after the workshop as well, if necessary, but it should be requested. Participants arriving earlier or leaving later should contact me for possible options, but housing is not guaranteed.

Accommodations and arrival: When you arrive, you need to check in at the Area Desk of the Living Learning Center, which is on 15th Avenue between University Street and Agate Street; there you will receive a key to your room. The confusing thing is that the building is disconnected (there is a South component and a North component). The component on 15th Avenue is the South component, but the component that contains the Area Desk is the North component (which is not bordered by any street at all). Also, if you arrive after 8pm, you will have to call this number: (541) 346-5686. When you depart, you will need to return your key to the Area Desk as well.

Transportation costs: Unless you have been told otherwise, we will not be covering any transportation costs.

Flying to Eugene: You should fly to the Eugene airport, code EUG. There are direct flights connecting Eugene to Seattle, Portland, Salt Lake City, Denver, San Francisco, and Los Angeles. In my personal experience, if you fly through San Francisco then your flight will be delayed and you will miss your connection, so try to find another option if possible. It is also possible to fly to Portland, which is a little over a two hour drive from Eugene - there are also shuttles and a train which can get you from Portland to Eugene, but this is typically far more complicated then just flying directly to Eugene (if you are being reimbursed and hope to choose this option, please contact me first, as price comparisons will need to be made).

Airport to Campus: There is no public transportation from the airport, so you'll have to take a cab, which should take about twenty minutes and cost about $30. Cabs in Eugene are run by a host of private companies, some better than others. Avoid Oregon Taxi if you can. Since a lot of you will be arriving around the same time, you might consider trying to find each other at the airport and sharing a cab. A google group will be set up closer to the time of the conference to help coordinate this. Cabs back to the airport will be organized later in the conference.