Foundations of Tropical Geometry

Workshop on Algebra and Representation Theory, Held on Oregonian Grounds

29 July - 2 August 2019
University of Oregon
Eugene, OR

Tropical geometry is a combinatorial shadow of algebraic geometry in which algebraic varieties are replaced by piecewise linear objects that record some information about the original variety. This has emerged as a major theme in 21st century algebraic geometry, with impact on moduli spaces, mirror symmetry, and connections to number theory, combinatorics, and many areas outside mathematics (such as phylogenetics, economics, and optimization).

In recent years there has been an increased focus on the foundations of the subject, and in particular on developing the tools of modern algebraic geometry in the tropical framework. This will allow more problems from algebraic geometry to be approached using tropical techniques.

The focus of this workshop will be the developing theory of tropical schemes, and the related ideas of blue schemes and tropical stacks. Along the way we will develop commutative algebra over semirings, and review some polyhedral geometry, and the combinatorics of matroids.

We will focus our attention on the following papers:
Equations of tropical varieties by Giansiracusa and Giansiracusa.
Tropical schemes, tropical cycles, and valuated matroids by Maclagan and Rincón.
Tropical ideals by Maclagan and Rincón.
Scheme theoretic tropicalization by Lorscheid.
The universal tropicalization and the Berkovich analytification by Giansiracusa and Giansiracusa.
A moduli stack of tropical curves by Cavalieri, Chan, Ulirsch, and Wise.

The workshop will be led jointly by Diane Maclagan and Jeff Giansiracusa, 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.

At a future date we will post a website with details on the background knowledge we expect from participants, and a guide for acquiring this background. WARTHOG workshops are meant to be broadly accessible.


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.

Logistics for Participants

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. They take away the coffee at 1PM. 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 a dormitory on campus. The cost of this will be covered by the workshop. Details will be emailed shortly before arrival.

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 Ben Elias for options, but housing is not guaranteed.

Transportation costs: Unless you have been told otherwise, we will not be covering your transportation costs. If other options have not availed you, please email me (Ben Elias) and we can discuss options. (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, 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 more complicated than just flying directly to Eugene.

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: Muscle Arm Taxi, ECabs, Eugene City Hybrid Taxi (more expensive), GOTaxi, Emerald city cab, these have all been good to me. If they ask 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 to help coordinate this. 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 very cheap.