Instructor: Alexander Dunlap (he/him; alexander decimal dunlap at duke decimal edu). Office: Physics 297. I will not have formal office hours for the minicourse but will be happy to discuss at any time I am around, or feel free to make an appointment.
Meetings: Mondays and Wednesdays, 10:05–11:20am, Gross 318. The course will also be streamed on Zoom; please contact the instructor for the link. The first meeting will be Wednesday, January 7 and the last meeting will be Wednesday, February 4. There will be no meeting on Monday, January 19 due to Martin Luther King, Jr., Day. There will be a makeup meeting on Friday, January 16 in Physics 235 (not the usual room).
Course description: The theory of regularity structures, introduced by Martin Hairer in the early 2010s, is a powerful tool for developing robust pathwise solution theories to nonlinear singular stochastic partial differential equations. It can be thought of as a far-reaching generalization of Taylor expansions, where the terms in the local expansion are no longer required to be polynomials but can additionally feature terms built from iterated stochastic integrals (or in principle even more exotic objects). This theory allows one to lift an a priori ill-posed PDE to a richer space, equipped with additional stochastic information, in which a necessary renormalization can be understood, and then in which the equation can be solved via a fixed-point argument. In this course I will introduce regularity structures and their use in the solution theory of the KPZ equation (a Hamilton-Jacobi equation with rough random forcing).
References:
Credit: During each lecture, I will specify one or more "DO" exercises. If you are enrolled in the minicourse, each Monday you need to turn in a solution to one of the "DO" exercises posed the previous week to receive credit for the course. At least one of the exercises each week should be straightforward, so this is not intended to be an onerous requirement. Everyone is welcome to attend the minicourse without enrolling as well, and then of course you don't need to turn in anything, but I still encourage you to give the exercises a try. Regularity structures is quite a technical subject and so it will really help in understanding to get a little hands-on practice. Graduate students are strongly encouraged to enroll.
(very much subject to change, but I will try to keep this table up-to-date)
| Meetings | Topic | Notes/References |
|---|---|---|
| W 1/7 | Brief introduction to the KPZ equation, Cole–Hopf transform, sketch of mathematical challenges. | Notes |
| M 1/12 | Young integral, motivation for rough paths via Itô integration, rough paths, controlled rough paths. | Notes. [FH] §2.1, §4.1–3 (to be continued on Weds.). |
| W 1/14 | Integration against controlled rough paths, sewing lemma. | [FH] §4.2–3. |