Bingxiao LIU : Lecture 2025 in UzK

 Title: Introduction to the Atiyah-Singer index theory (click here to see a lecture introduction).

Information:

• We will also have lecture on Monday 2^nd June (10h00–11h30, Room 204).

• Tutorial Classes on 16^th and 23^rd June, but there will be no lectures on 18^th, 19^th, 25^th, and 26^th June.

• Tutorial Class on 12^th May, but there will be no lectures on 14^th and 15^th May.

• The first lecture will be on Wednesday 16^th April 2025.

• The first tutorial class (exercise session) will be on Monday 28^th April 2025 (10h00–11h30, Room 204).

• Tutor for Exercise Classes: Chin-Chia Chang

• Exercise classes will be held on the following Mondays (the schedule is regularly updated):

  • 28th April, 12th May, 26th May, 16th June, 23rd June (After this date: to be announced)

  • While you don’t have to hand in homework sheets, we highly recommend active participation in the exercise classes and preparation of the material beforehand. This is because the content of the exercises is beneficial for understanding the lecture material.

Lecture Timetable and Course Materials (Notes and homework sheets)

Texified lecture notes (regularly updated, last update: 08/06/2025) : IndexTheory2025UzK

Part 1: Introduction; Preliminary on differential geometry and differential operators on manifolds; Statements of Atiyah-Singer Index Theorem.

1. 16^th April: Index Theory 2025 Notes Lecture 1

2. 17^th April: Index Theory 2025 Notes Lecture 2

3. 23^rd April: Index Theory 2025 Notes Lecture 3

4. Homework sheet 1: Homework01

Part 2: Connection and curvature; Characteristic classes; Chern-Weil theory

1. 24^th April: Index Theory 2025 Notes Lecture 4

2. 30^th April: Index Theory 2025 Notes Lecture 5

3. 7^th May: Index Theory 2025 Notes Lecture 6

4. 8^th May: Index Theory 2025 Notes Lecture 7

5. No lectures on 14^thand 15^th May.

6. Homework sheet 2: Homework02

Part 3: Quillen’s superconnection and Chern character

1. 21^st May: Index Theory 2025 Notes Lecture 8

Part 4: Clifford algebra, spin group and spinors

1. 22^nd May: Index Theory 2025 Notes Lecture 9

2. 28^th May: Index Theory 2025 Notes Lecture 10

3. Homework sheet 3: Homework03

Part 5: Dirac operator, spin manifold and Lichnerowicz formula

1. 2^nd June: Index Theory 2025 Notes Lecture 11

2. 4^th June: Index Theory 2025 Notes Lecture 12

3. 5^th June: Index Theory 2025 Notes Lecture 13

4. Homework sheet 4: Homework04

5. Homework sheet 5: Homework05