MA323 – Topology, Spring 2025

  • Assignment 1 due Friday, Feb. 28: [Y] Sec. 1.1 #2, 3, 4, 5

  • A bit motivation in the Year of the Snake: 1, 2, 3 (check out other semesters' "motivation" sections for more applications and encounters with topology!)

Instructor

朱一飞 ZHU Yifei

CoS-M705

zhuyf@sustech.edu.cn

Office hours: Thursdays 10:20 am–12:10 pm and by appointment

Graders: 文子杰 WEN Zijie and 殷春双 YIN Chunshuang

Prerequisites

Abstract Algebra (MA214/219) or consent of the Department.

Objectives

This course introduces basic notions, examples, and applications of topology, a subject aimed to classify "spaces" according to those of their properties that are invariant under continous deformations (in contrast to those rigid ones adhering to metrics).

With the classification theorem for closed surfaces as a main goal, we will cover point-set topology systematically together with elements of geometric and algebraic topology. This includes categories, functors and their adjunction, universal properties, and examples; compactification; topological manifolds, partition of unity, and embeddings of manifolds into Euclidean spaces; simplicial complexes, Euler characteristic, and orientation; function spaces, the compact–open topology, homotopy, and basic ideas of higher categories; projective spaces, lens spaces, Dehn surgery, and knots; glimpses of topological classifications of quantum mechanical systems and of topology-enhanced machine learning.

References

The main references:
  • [Y] 尤承业，基础拓扑学讲义 (the main textbook, minimum requirement; good organization), Chapters 1–4
  • [M] James R. Munkres, Topology (a classic textbook; many in-depth, elaborate discussions and examples; somewhat old-fashioned; mostly optional reading on selected topics and exemplary mathematical exposition in general)
  • [BBT] Tai-Danae Bradley, Tyler Bryson, and John Terilla, Topology – A categorical approach (a recent textbook; supplementary with categorical perspectives)
Some additional references:

Exams

There will be one in-class midterm exam, on April 18, and one final exam. Each of these exams is worth 30% of your final grade.

Homework

The assigned problems for each week are due each Friday in-class at 2 pm, listed at the top of this page. Homework is worth 40% of your final grade. Students must make arrangements in advance if they will not be handing in homework on time. You may send an electronic version of your homework (TeX'd up or scanned properly, in PDF format) to the graders at the mouse-over email addresses from above.

We encourage you to discuss homework problems with your classmates, including strategies for solving different kinds of problems. However, when you actually write up your solutions, you must do this on your own.