These lectures will introduce the geometry of multigraded syzygies. Classically, the geometry of syzygies refers to using free resolutions over a standard graded polynomial ring to study projective geometry, via techniques pioneered by Ein, Eisenbud, Green, Lazarsfeld, Voisin and many others. Our focus will be on using free resolutions over multigraded polynomial rings to give new insights into toric geometry.
The central theme of our lecture series is how theorems in the standard graded case have analogues in the multigraded case, but finding these analogues often requires creative new perspectives on the classical results. We will demonstrate this theme by exploring three famous theorems about syzygies, and their recent multigraded analogues: Hilbert's Syzygy Theorem, Beilinson's resolution of the diagonal for projective space, and Mark Green's Np Theorem for curves.
The workshop will be led by Michael Brown and Daniel Erman, and will consist of a combination of lectures and problem sessions.
It will be co-organized by Ben Elias and Andrew Hanlon, under the watchful eye of Nicholas Proudfoot. In 2027 it is supported by NSF grant DMS-2039316.
If you are interested in participating, email Andrew Hanlon. 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. If space or funding becomes limited, priority will typically be given to graduate students over postdocs. 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.
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 breakfast 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 Andrew.
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 be in contact 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 Andrew, and we can discuss options. The amount of available funding is uncertain, as this event will be closing out our grant. If we do end up covering your transportation, it may be better for us to book the travel ourselves, so please be in contact before making any purchase.
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 is more likely to be delayed leaading to a missed connection. 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.