These are the lecture slides for my lecture Maunderings in enumerative geometry at UC Santa Cruz on Friday afternoon.
I’m going to be speaking about “Spherical Varieties and Tropical Geometry” in the Invitation to Mathematics seminar in two successive weeks:
- Wednesday, October 29, 2014 – 4:10pm to 5:10pm in Cockins Hall 218
- Wednesday, November 5, 2014 – 4:10pm to 5:10pm in Cockins Hall 218
Here’s my abstract:
This is intended as a prospectus of research, and I’m looking for students who want to work on this project. A complex algebraic variety is called a spherical variety if it’s acted upon by a reductive group and there is a dense orbit under the action of a Borel subgroup. To begin, I will explain some generalities about spherical varieties and the convex bodies associated to them, mostly via examples. Then I will focus on the research of Jason Miller, some of which appears in his 2014 Ohio State dissertation. Then, making a fresh start, I will say something about tropical varieties, which are piecewise linear or skeletal versions of algebraic varieties; again there will be examples. Tropical varieties naturally live in the world of toric varieties, which are extremely special examples of spherical varieties. The goal of the project is to find notions akin to those of tropical geometry in the wider world of spherical varieties.
For a related reading list, see my earlier post.
Here’s a link to the 2014 Ph.D. thesis of Jason Miller, Okounkov Bodies of Borel Orbit Closures in Wonderful Group Compactifications.
This is a reading and viewing list for a small group of people who are working to learn these topics. Anyone is welcome to use it, of course.
- Algebraic groups and representation theory
- Humphreys, Linear Algebraic Groups (Springer)
- Toric varieties
- Cox, Little, Schenck, Toric varieties (AMS)
- Tropical geometry
- Spherical varieties
- Associated convex bodies
- Lazarsfeld and Mustaţă, Convex Bodies Associated to Linear Series, (arXiv:0805.4559)
- Kaveh and Khovanskii, Convex bodies associated to actions of reductive groups, (arXiv:1001.4830)
- Kiritchenko, Smirnov, Timorin, Schubert calculus and Gelfand-Zetlin polytopes, (arXiv: 1101.0278)
- Miller, Okounkov Bodies of Borel Orbit Closures in Wonderful Group Compactifications, Ph.D. dissertation at the Ohio State University, 2014
- Videos from the workshop Convex Bodies and Representation Theory at the Banff International Research Station in February 2014