Image of an excited mathematician by Ryan Armand
Instructor: Mike Miller
Email: smm2344@columbia.edu
Webpage: here! homework will also be posted to Courseworks
Office: My dining room table, most likely (UPDATE: my desk is now in my bedroom instead)
Office hours: There are two Zoom office hours per week; one is Friday 12-1PM and another is Monday 3-4PM (both EST). If you can't make it to either of these but have questions you want to talk about, just get in touch with me!
Teaching assistant: Anda Tenie (ast2175@columbia.edu) is available for questions on the Discord, and holds office hours Monday 6-8PM (EST), at this link: https://columbiauniversity.zoom.us/j/95296029860
Topics: The course is separated into two halves: first, point-set topology (essentially, the fundamental tools of topology which are commonly used here and elsewhere in math); second, algebraic/geometric topology, where we apply the tools from the first half of the course to situations where we can actually see a picture of what's going on.
Point-set topology
Refresher on metric spaces. Open sets, topologies, and continuous maps.
Basic operations on sets in topological spaces.
Homeomorphisms.
Products of topological spaces.
Connectedness, path-connectedness.
Hausdorff spaces.
Compactness in its various guises.
Quotient spaces.
Algebraic and geometric topology
Groups and presentations.
The fundamental group.
The Brouwer fixed point theorem.
Classification of compact surfaces.
The Jordan curve theorem, if time permits.
Textbooks: We will be eclectic in our choice of references.
The first half will reasonably closely follow Hatcher's notes on point-set topology, though if time permits we'll talk about some topics that he doesn't write about.
References for the second half will be more scattered. Our discussion of fundamental groups will refer to portions of Chapter 1 of Hatcher's algebraic topology book. The proof of the Jordan curve theorem will follow Maehara's argument. And the classification proof will be idiosyncratic, not really following any one author, though you might also like Conway's ZIP proof. (Ours will use technology we've built that Conway's argument --- intended to use as little as possible --- doesn't have access to, so doesn't use.)
If you want additional references for your personal use, Munkres' book is usually the standard reference, and quite comprehensive; some additional topics in the first half that aren't covered in Hatcher will certainly be covered here.
Homework: There will be a total of ten homework assignments. Homework will be assigned on Tuesday and due the following Tuesday. To submit your homework, you need to upload it on Courseworks. You have two options: write neatly, and scan using a good scanner (or scanner app on your phone / tablet), or learn to TeX your homework. If your homework is not readable, it will not be accepted.
I encourage working with your classmates on your homework, but your submissions must be your own work. You can work on ideas and problems with each other, but ultimately, you should submit something where you have written down your own thought process / argument in your own words. Copying homework solutions, from classmates or online, is considered academic dishonesty and will be treated as such.
Your lowest homework score will be dropped.
Late homework will not be accepted.
Discord: This semester there will be a class Discord channel, where you can talk to your classmates about the material, the homework, or ask me questions. I'm imagining this like a less clunky Piazza, that hopefully people already have installed.
I will direct that most math questions you ask me be posted to the Discord, so that my response is a matter of the public record (instead of stuck on an email chain between the two of us, that nobody else gets to see.) I will respond to questions there.
The link to the Discord will be posted in a Courseworks announcement. You will be asked to post your UNI so I can cross-reference. Only one Discord account per UNI will be allowed into the server.
Tests: There will be one midterm exam (set to cover material from the first half of the course) and one final exam (cumulative). They will both be take-home exams, scheduled for one full day; you will have the full day to work on them.
Midterm: October 25 (12AM-11:59PM)
Final: Tentatively, December 17
Project: Students this semester had the choice between a final project (an expository paper on a topology topic of their choice) and a final exam. Some students have chosen to make their projects publicly available here.
Grading: The final course grade is weighted as:
Homework: 40%
Midterm: 25%
Final: 35%
Your bottom homework score will automatically be dropped.
Students with disabilities: To receive accomodations for exams (or otherwise), you must register with the Disability Services office and present an accomodation letter.
Image of an excited mathematician by Ryan Armand