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)

2. 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.

3.  Scheme

Homework 50% plus in-class final presentations 50%

4. 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.

5. Reference material (textbooks)

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

6. Credit 2

7. Course Number/ ID

No.: NCTS 5055 (三校聯盟之學生於課程網選課適用)

ID: V41 U3011

8. Join the course online

TBA

9. Course Video

TBA