MA323 – Topology, Spring 2024

  • The final exam will take place on Monday, June 17, 10:30 am–12:30 pm, in Lecture Hall 3-303.

  • Office hours during the final exam weeks: Thursday, June 13, 10:20 am–12:10 pm.

  • Here is the midterm exam.

  • Assignment 12, required but not collected: Sec. 4.3 #3; Sec. 4.4 #4, 8, 10, 11, 13; Sec. 4.5 #2, 3, 5, 8, 10.
    Reading: [Y] Secs. 4.3–4.5, Appendix A; [BBT] Sec 6.7; [M] Sec. 55.
    Optional reading: [Y] Appendix B; [M] Secs. 67–69 categorically, 56.

  • Assignment 11 due Tuesday, May 28: [Y] Sec. 4.2 #4, 5, 6; [M] Sec. 52 #7.
    Reading: [Y] Sec. 4.2; [BBT] Sec 6.2.

  • Assignment 10 due Tuesday, May 21: [Y] Sec. 4.1 #2, 3, 4, 5, 6, 7, 8.
    Reading: [M] Secs. 36, 46; [BBT] Secs. 5.1, 5.6.1, 6.1; [Y] Sec. 4.1.
    Čech complexes (p. 51 onwards) applied in topological data analysis

  • Assignment 9 due Tuesday, May 7.
    Reading: [Y] Secs. 3.3, 3.4; [B] Secs. 3.2, 3.4, 3.5.
    Visualization of adding handles
    Computational topology: Computational aspects of surfaces can be found in Chapter II.
    TOP vs DIFF vs PL
    Optional reading: [B] Sec. 3.6.

  • Midterm exam on Friday, Apr. 19 covers [Y] Secs. 1.1–1.3, 2.1–2.6, and 3.1–3.2.

  • Assignment 8 due Tuesday, Apr. 16: [Y] Sec. 2.5 #1, 5, 6, Sec. 2.6 #1, 3, 4, 5, Sec. 3.2 #8, 9, 14.
    Reading: [Y] Secs. 2.5, 2.6, 3.1, 3.2; [BBT] Sec. 1.3.
    Klein bottle as a distribution space for image data: discussion begins on page 96 of slides; and as reflected from the topology of convolutional neural networks.
    Hopf maps: Visualization of $S^3\to S^2$; Hopf "bundles" as eigenbundles for Hamiltonians in quantum mechanics (Section 5 here, visualization over $\mathbb R$).

  • Assignment 7 due Tuesday, Apr. 9: [Y] Sec. 2.4 #4, 5, 8, 10.
    Reading: [Y] Sec. 2.4.
    Optional reading: [M] Sec. 24.

  • Assignment 6 due Tuesday, Apr. 2: [Y] Sec. 2.3 #1, 2, 4, 5, 6, 7, 9, 11, 16, 18, 19.
    Reading: [Y] Sec. 2.3; [BBT] Sec. 5.5.
    Examples of compact spaces: higher-dimensional spheres Optional reading: [M] Secs. 27, 37.

  • Assignment 5 due Tuesday, Mar. 26: [Y] Sec. 2.2 #1, 2, 3, 4; [M] Sec. 33 #3, Sec. 35 #7b.
    Reading: [Y] Sec. 2.2; [M] Secs. 33–35.

  • Assignment 4 due Tuesday, Mar. 19: [Y] Sec. 2.1 #5, 6, 12, 14, 16, 18.
    Reading: [Y] Sec. 2.1; [M] Secs. 30–32.

  • Assignment 3 due Tuesday, Mar. 12: [Y] Sec. 1.3 #3, 4, 6, 8, 11.
    Reading: [Y] Sec. 1.3; [BBT] Secs. 1.4, 0.3.3, 0.3.4.
    Optional reading: The structure of meaning in language: Parallel narratives in linear algebra and category theory (interesting applications to Large Language Models begin in the section "Structures in the Real World" on page 6, with Figure 1, but need Isbell adjunction introduced in the previous section in parallel with Singular Value Decomposition for Principal Component Analysis)

  • Assignment 2 due Tuesday, Mar. 5: [Y] Sec. 1.2 #2 (also show that the subspace topology is in fact the coarsest topology on $B$ for which the inclusion $i$ is continuous), 4, 5, 6, 9, 10, 12, 13.
    Reading: [Y] Sec. 1.2; [BBT] Secs. 0.2.1, 0.2.2.

  • Assignment 1 due Tuesday, Feb. 27: [Y] Sec. 1.1 #3, 7, 14, 6, 8, 13, 15.
    Reading: [Y] Introduction (set operations, map, cartesian product, equivalence relation), Sec. 1.1.
    Optional reading: [M] Secs. 20, 21, Exe. 20.8.

Instructor

朱一飞 ZHU Yifei

CoS-M705

zhuyf@sustech.edu.cn

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

Grader: 郑力豪 ZHENG Lihao

Prerequisites

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

Although strictly (and traditionally) speaking notions such as groups and group homomorphisms will not be used until the latter half (or even later) of this course on the fundamental group(oid) of a topological space and its computation, familiarity with basic abstract algebra is necessary for acquiring a sophistication with writing proofs and for supplying examples, intuitions, and analogies.

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, incorporating standard Chinese textbooks such as 尤承业《基础拓扑学讲义》, Chapters 1–4, as well as Munkres's Topology and Bradley, Bryson, Terilla's Topology – A categorical approach (2020). 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 physical systems and of topology-enhanced machine learning.

References

[Y] 尤承业，基础拓扑学讲义，北京大学出版社，1997. (The main textbook, minimum requirement; good organization.)

[M] James R. Munkres, Topology, Prentice Hall, Inc., Upper Saddle River, NJ, 2000, second edition of [MR0464128]. (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, MIT Press, Cambridge, MA, 2020. MR4232168 (A new textbook; supplementary with categorical perspectives.)

[B] 包志强，点集拓扑与代数拓扑引论，北京大学出版社， 2013. (A relatively recent textbook; supplementary with selected topics.)

[A] Mark Anthony Armstrong, Basic topology, Undergraduate Texts in Mathematics, Springer-Verlag, New York-Berlin, 1983, corrected reprint of the 1979 original. MR705632 (For additional perspectives.)

Exams

There will be one in-class midterm exam, on April 19, 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 Tuesday in-class at 2 pm, listed on 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 grader at the mouse-over email address 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.