2025/2/20-2025/6/12  

Every Thursday, 10:20-11:10, 11:20-12:10

Room 509, Cosmology Building, NTU 

Outline

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

1. Introduction and Contents

(I) MOTIVATION

- the idea of homology and cohomology

(cycles in a manifold)

- the idea of quantum cohomology

(cycles in a space of mappings)

- Examples of quantum differential equations

(commutativity versus noncommutativity)

(II) DIFFERENTIAL EQUATIONS AND DIFFERENTIAL GEOMETRY

- o.d.e. in the complex plane or Riemann sphere

(canonical solutions from the Frobenius Method)

- the Stokes Phenomenon

(canonical solutions in sectors)

- flat connections, parallel translation

(multivalued flat sections)

- the fundamental group and monodromy

- the idea of integrable systems

(zero curvature equations)

- the Painleve property and isomonodromy equations

- the harmonic map equation

(harmonic maps into symmetric spaces)

(III) ADVANCED TOPICS

- the DPW method

(loop group method)

- the idea of the naHC

(nonabelian Hodge Correspondence)

- the idea of topological-antitopological fusion

(conformal field theory)

- examples of the tt* equations

(tt*-Toda equations)

- Stokes data of the tt*-Toda equations

(towards algebraic and categorical ideas)

Course Goal

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.

Prerequisites 

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.

Credit

2

Scheme

Homework 50% plus in-class final presentations 50%

Course Number/ ID

No.: NCTS 5055
ID: V41 U3012
(三校聯盟之學生於課程網選課適用)

Course Online Meeting Link

Course Video

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

Lecture note

TBA