MAT7064 – Topics in Geometry and Topology, Fall 2024

• Assignment 4 due Friday, Oct. 18


• Assignment 3 due Friday, Oct. 11


• Assignment 2 due Friday, Sep. 27


• Assignment 1 due Friday, Sep. 20

Instructor

朱一飞 ZHU Yifei

CoS-M705

zhuyf@sustech.edu.cn

Office hours: Official times are Thursdays 10:20 am–12:10 pm for questions about the course material or geometry and topology in general. If you have more immediate concerns or cannot come during ordinary office hours, feel free to contact me by email or in person.

Grader: 孙运毫 SUN Yunhao

Objectives

Algebraic topology, especially in the form of homotopical and categorical methods, plays an increasingly vital role in many of today's central areas of study in mathematics and science, including number theory, data science, and condensed matter physics. This iteration of the topics course will cover homotopy theory roughly corresponding to Chapter 4 of Hatcher (see below), along with supplementary materials such as fiber bundles, characteristic classes, spectral sequences, rational homotopy theory, cohomology operations, bordism, and Thom spaces. It also serves as the second half of a one-year course in modern algebraic topology, following MAT8021, though the order can be switched.

Prerequisites

MAT8021 Algebraic Topology is not required. Indeed, homotopy theory is relatively independent of homology and cohomology (see here, for example). However, as the course progresses, it will be helpful to incorporate the latter into the discussion, even if taking them as blackboxes, such as when we reach spectral sequences.

Textbook

The main references will be Hatcher's Algebraic Topology (online version available here along with additional exercises and content items).

Other useful references include Haynes Miller's Lectures on algebraic topology (2022).

Homework

There will be weekly problem sets to be handed in during class each Friday. The first problem set is due Friday, September 20. Your lowest problem set score will be dropped from your final grade. Homework will be posted online on the course webpage.

Late homework will be docked by 15% per day (or portion thereof) up to a maximum of 45%, unless you have made prior arrangements.

You are allowed (and encouraged) to work with other students while trying to understand the homework problems. However, the homework that you hand in should be your work alone.

Exams

There will be none.