For many (affine) algebraic varieties appearing in geometric representation theory and algebraic combinatorics, their coordinate algebra admits the structure of a cluster algebra. These algebraic varieties include (in order of increasing generality) the basic affine space, (double) Bruhat cells, positroid varieties, Richardson varieties, and braid varieties. Roughly speaking, this means that the corresponding algebraic variety comes with a collection of coordinate toric charts, with explicit coordinates and transformation rules between the charts that can be codified via quiver combinatorics. Conversely, the quiver and the coordinates can be read from combinatorial and geometric properties of the variety itself, though this is more of an art than a science.
The goal of this workshop will be to understand the combinatorics and geometry underlying these cluster structures in the case of braid varieties, which include all of the cases described above and was established quite recently. The combinatorics can be encoded in two very different objects: weaves and 3D plabic graphs. Both appear in a variety of mathematical settings, including Soergel calculus and connections to high-energy physics. These rich combinatorial objects and their ties to other areas of mathematics inspire many questions about braid varieties, their cluster structures, and beyond.
The workshop will be led by Eugene Gorsky and Melissa Sherman-Bennett and José Simental Rodríguez, 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. In 2025 it is supported by NSF grant DMS-2039316.
If you are interested in participating, email Ben Elias. Use [WARTHOG] Registration as your subject please. Please include your school, advisor, and a brief description of your research interests. Funding for accommodations in Eugene (but not for travel) will be available to students and postdocs as long as space remains. Limited funds are available for travel upon request, primarily for US citizens or permanent residents.
This is not a final confirmation of attendance, which will be requested at a later date.
In the near future we will provide a webpage describing the background we expect you to acquire before the workshop begins. Typically this is what one expects from a graduate-level algebra course, with a few extra bits which can be acquired with some pre-workshop study, and we give a roadmap for acquiring that knowledge.
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 (+ salads) party at a local park, sponsored by the workshop.
Our typical daily routine is: hour of lecture, 45 minutes of exercise, 15 minute break, repeat four times, with a longer break for lunch.
Food: Starting at 830AM each day of the workshop, coffee and bagels will be catered to the lecture room. Aside from this and the pizza party, participants will have to pay for their own food. Several nearby residence halls have a cafeteria with more substantial breakfast for purchase, and there are many restaurants in the EMU (student center). There will be time to eat lunch at many local restaurants.
Accommodations: Participants will be staying in a dormitory on campus, in single occupancy rooms which have both shared bathrooms and single-occupancy gender-neutral bathrooms. Linens and towels are provided. The cost of this will be covered by the workshop. Details will be emailed shortly before arrival. Participants with special accommodation needs are encouraged to email Ben.
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. Participants arriving earlier or leaving later should contact me for possible options, but housing is not guaranteed.
Transportation costs: Unless you have been told otherwise, we will not be covering your transportation costs. However, if other options have not availed you, please do email me (Ben Elias) and we can discuss options. We have funds to cover travel for a decent number of participants, especially US citizens, but not enough for everyone. If we do end up covering your transportation, make sure to fly to EUG during the expected dates, or contact me to discuss price comparisons.
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, Los Angeles, and recently Chicago. 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 shuttles and a train which can get you from Portland to Eugene, but this is typically more complicated then just flying directly to Eugene.
Airport to Campus: There is no public transportation from the airport, so you'll have to take a cab or rideshare, which should take about twenty minutes and cost about $30-$40. Cabs in Eugene are run by a host of private companies, some better than others. Avoid Oregon Taxi if you can. If the driver asks which route to take, just say to take 99. 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, which helps with coordination. Cabs back to the airport will be organized later in the conference.
Getting around: During the conference you will be able to walk everywhere you need to go. If you are staying longer in Eugene or in Oregon for recreational purposes, it is easy to rent a bicycle, and car rentals (within the city, not from the airport) are decently priced.