This workshop is part of RIMS Research Project 2020 "Differential Geometry and Integrable Systems - Mathematics of Symmetry, Stability and Moduli -"

Speaker:

Eric Rains (Caltech)

Title:

The (noncommutative) geometry of difference equations

Abstract:

Many important special functions either satisfy nice differential or difference equations (hypergeometric functions) or describe nice flows in families of such equations (Painlevé, Garnier, etc.). This leads to a pair of natural problems: (1) How can one classify equations with given singularities? (E.g., when is the equation unique? This holds for the main hypergeometric cases.) (2) What are the isomorphisms between these "moduli spaces"? (E.g., canonical isomonodromy deformations.) Although there are well-known approaches to such problems coming from algebraic geometry, they can be quite difficult to apply in practice, and thus ideally one would reduce to previously solved instances. This can (mostly) be done here, with the key idea being that there is a recipe for turning differen(ce/tial) equations into sheaves on noncommutative projective surfaces. The result is that many natural questions about the former reduce to questions about the latter, and in many cases can even be reduced to the commutative case, letting us apply classical algebraic geometry. I'll try to give a flavor of how the noncommutative approach works, and then discuss in more detail how the commutative relaxation applies. (E.g., given any differen(ce/tial) equation, there is a recipe for classifying its generalizations and special cases.)

Dates: May 18th (Mon.)--29(Fri.), 2020.

postponed to November 2021.

15th Mon. 4-6 pm PST = (UTC -8)/ 16th Tues. 9-11 am JST = (UTC +9)

16th Tues. 4-6 pm PST/ 17th Wed. 9-11 am JST

17th Wed. 4-6 pm PST/ 18th Thurs. 9-11 am JST

18th Thurs. 4-6 pm PST/ 19th Fri. 9-11 am JST

23rd Tues. 4-6 pm PST/ 24th Wed. 9-11 am JST

24th Wed. 4-6 pm PST/ 25th Thurs. 9-11 am JST

At least 30 minutes per day is devoted to questions.

Venue:

Zoom

Registration:

Registeration is closed.

Notes: Day#1 Day#2 Day#3 Day#4 Day#5 Day#6

Videos: Day#1 Day#2 Day#3 Day#4 Day#5 Day#6

Main references:

Eric Rains, Generalized Hitchin systems on rational surfaces, arXiv:1307.4033.

Eric Rains, Birational morphisms and Poisson moduli spaces, arXiv:1307.4032.

Eric Rains, The birational geometry of noncommutative surfaces, arXiv:1907.11301

Related overview lecture:

Video, Slides

Organizers:

Yoshihiro Ohnita (Chair, Osaka City University & OCAMI)

Kazuki Hiroe (Chiba University)

Akane Nakamura (Josai University)