ncts

0. Introduction to the February-June 2025 course

The course will focus on part (III) "Advanced Topics" of the syllabus below. The introductory material covered in

the previous semester is not a pre-requisite for this semester, but may be useful as motivation. Some lecture notes from the previous semester are available here:

https://ncts.ntu.edu.tw/events_3_detail.php?nid=344

This semester Lie theory (finite and infinite dimensional) will be used

systematically. This will provide more powerful tools, a clearer conceptual

framework, and clarify links with other parts of geometry. The main topics will be:

- preliminaries on connections and integrable systems compatible with a Lie group G

- harmonic maps and harmonic bundles

- the holomorphic/harmonic correspondence (DPW Method, loop group method, non-abelian Hodge

Correspondence)

- Stokes data of the quantum differential equations and the tt* equations

- algebraic aspects of the Stokes data of the quantum differential equations and the tt* equations

No special knowledge will be assumed, just linear algebra, basic topology, and ordinary differential equations. However, some familiarity with differentiable manifolds and Lie groups will be useful.

The purpose of this course is to introduce students to several important topics in geometry, topology, and integrable systems theory. The goal is to reach some important mathematical problems motivated by the physics of conformal field theory. At the same time the course will involve some very classical mathematics, such as special functions and the Stokes Phenomenon. Most of all, the course will demonstrate how different areas of mathematics can interact and lead to interesting problems.

There is no textbook, but references and reading material will be given during the lectures.

Since the course videos are not publicly available, please contact Ms. Murphy at <murphyyu@ncts.tw> to request access for viewing.