Course format

This class will be offered entirely online. We will meet live during class time over Zoom, where (on top of traditional lecturing activities) we will engage in group work and you will be able to ask questions. To the extent possible, instructor will upload lecture videos after every class. Office hours will take place online.

If you are taking this class, but have any concerns about accessibility--including to fast-enough internet, to technology (such as a laptop), and to a quiet location where you can attend lecture--let Hiro know as soon as possible.

Tu/Th: 6:30 PM - 7:50 PM.

Course description

This class assumes you have taken a course in undergraduate point-set topology (so that you are familiar with topological spaces, continuity, compactness, metric spaces, connectedness, and Hausdorffness).

In our class, you will develop further fluency with the above topics. Another goal of this class is for you to become familiar with mathematical thinking---questioning and understanding why definitions exist, identifying when you or another communicator is being precise or imprecise (and for what purpose), developing tastes that are rooted in practice and informed experience, exploring the mathematical landscape on your own.

Prerequisite: MATH 4330.

Textbook and resources.

You do not need to buy a textbook for this class. The following are freely available resources on point-set topology:

1. Here is a link to the Fall 2019 and Fall 2020 websites for Math 4330. There you will find necessary review of point-set topological ideas.
2. Allen Hatcher's notes on point-set topology.
3. Stefan Waner's notes on Elementary Topology.
4. Sidney A. Morris's Topology without Tears.

Resources for proof (all freely available):

1. Book of Proof by Richard Hammack. You will need to be comfortable with all the material in this book.
2. Introduction to Proof in Analysis by Steve Halperin. You will need to be comfortable with Chapters 1-3 of this book.
3. Introduction to mathematical arguments by Michael Hutchings. You will need to be comfortable with all this material.

The survey

Here is the survey. Please fill out by Thursday, January 21, at 11:59 PM. It should take no more than 45 minutes. Names will be removed from the survey responses, but all other results of the survey will be shared with the class.

Important dates

Final Paper Topic Proposal Due Date: Fri Mar 12. (Note the date change; it is also no longer a midterm paper.)
Final Paper Draft Due Date: Wed Apr 21.
Final Paper Due Date: Wed May 12.

Collaboration policy

I strongly encourage all of you to collaborate. Please do so. If you do, you must indicate clearly on every assignment that you have collaborated, and indicate with whom. You may also explore resources online, but if you show do not show a full understanding of the math you borrow from other sources, I will be quite harsh in grading accordingly. You must also cite all sources. You will get a zero on your assignment if you do not cite something that needs citation; you will also get a negative score for repeat offenses.

Finally, you must write solutions on your own. It is fine to think through problems and find solutions with each other, but when it comes to the act of writing your homework, you must do so without assistance from another. This is because the act of solving something and writing a mathematical proof are two different skills, and I want you to also hone the latter. As an extreme anti-example, copying and pasting solutions/proofs will not be tolerated. To reiterate, you may not write solutions together.

Recordings of Zoom Lectures

Can be found here.

Notes

1. Tue, Jan 19. Discussion questions.
2. Thu, Jan 21. Topology and metric on R^4.
3. Tue, Jan 26. Metrics on 2-by-2 matrices.
4. Thu, Jan 28. Orthogonal matrices.
5. Tue, Feb 2. S^1 x S^1.
6. Thu, Feb 4. RP^1 and S^1 x S^1.
7. Tue, Feb 9. RP^2.
8. Thu, Feb 11. RP^2 and contractible loops. Notes were not saved; see Zoom recording at MediaFlo.
   Tue, Feb 16. Snowpacolypse.
   Thu, Feb 18. Snowpacolypse.
   Tue, Feb 23. Snowpacolypse.
9. Thu, Feb 25. The fundamental group.
10. Tue, Mar 2. Groups and pi1.
11. Thu, Mar 4. Homotopy equivalence and group actions. Notes were not saved; see Zoom recording at MediaFlo.
12. Tue, Mar 9. Group actions. Notes were not saved; see Zoom recording at MediaFlo.
13. Thu, Mar 11. Euler characteristic and higher homotopy groups.
14. Tue, Mar 23. Some discussion of writing assignments; the four-color theorem; dimension. Notes were not saved; see Zoom recording at MediaFlo.
15. Thu, Mar 25. Dimension and simplicial complexes.
16. Tue, Mar 30. Simplicial complexes and CW complexes.
17. Thu, Apr 1. CW complexes.
18. Tue, Apr 6. Euler characteristic and more examples of CW complexes.
19. Thu, Apr 8. More Euler characteristic, more CW complexes, and RPn, RPoo.
20. Tue, Apr 13. Homology (mod 2) of CW complexes
21. Thu, Apr 15. Answering some questions; homology of 0-dimensional things.
22. Tue, Apr 20. Homology of spheres.
23. Thu, Apr 22. Twenty Questions.
24. Tue, Apr 27. Practice for comprehensive exams.
25. Thu, Apr 29. Last day of classes. What's next? Topological data analysis and other topics. Practice for comprehensive exams.

Homeworks

Make sure to fill out the survey by Thursday, January 21, at 11:59 PM.
All assignments are due on Canvas, and must be uploaded in PDF format. All true/false questions are also to be answered at the corresponding quiz on Canvas.
All homework assignments will be shared with your classmates, so if you wish, you may remove your names from your scanned/typed/written assignments. (When you upload your homework, I will know which assignments belong to whom, thanks to Canvas.)

1. Extra Credit 1. Due Friday, January 22, 11:59 PM.
   Writing 1. Due Monday, January 25, 11:59 PM.
   Journal entry due. Due Wednesday, January 27, 11:59 PM.
2. Extra Credit 2. Due Friday, January 29, 11:59 PM.
   Writing 2. Due Monday, February 1, 11:59 PM.
3. Extra Credit 3. Due Friday, February 5, 11:59 PM.
   Writing 3. Due Monday, February 8, 11:59 PM.
   Journal entry due. Due Wednesday, February 10, 11:59 PM.
4. Extra Credit 4. Due Friday, February 12, 11:59 PM.
   Writing 4. Due FRIDAY, FEBRUARY 26, 11:59 PM.
   Journal entry due. Due WEDNESDAY, MARCH 3, 11:59 PM.
   Final Paper Topic Proposal. Fri Mar 12. (Note the date change; it is also no longer a midterm paper.)
5. Extra Credit 5. Due Friday, March 12 11:59 PM.
   Writing 5. Due Monday, March 15, 11:59 PM.
6. Extra Credit 6. Due Friday, March 26 11:59 PM.
   Writing 6. Due Monday, March 29, 11:59 PM.
   Journal entry due. Due Wednesday, March 31, 11:59 PM.
7. Extra Credit 7. Due Friday, April 2, 11:59 PM.
   Writing 7. Due Monday, April 5, 11:59 PM.
8. Extra Credit 8. Due Friday, April 9, 11:59 PM.
   Writing 8. Due Monday, April 12, 11:59 PM.
   Journal entry due. Due Wednesday, April 14, 11:59 PM.
9. Extra Credit 9. Due Friday, April 16, 11:59 PM.
   Writing 9. Due Monday, April 19, 11:59 PM.
10. Extra Credit 10. Due Friday, April 23, 11:59 PM.
    Writing 10. Due Monday, April 26, 11:59 PM.