Topology is the study of topological spaces, continuous maps between them, and properties preserved under continuous transformations. Topological spaces are ubiquitous in mathematics; Banach spaces, metric spaces, and varieties are all types of topological spaces.
In this course, we will start by introducing metric and general topological spaces, including basic properties like compactness and connectedness. We will then study elements of algebraic topology, including fundamental groups and covering spaces, homotopy and the degree of maps.
There will be two midterm exams in the course - the first exam will be in October and will be an in-class written exam. The second exam will be in November and will be an individual oral exam. The week before the exam, I will provide a list of topics and questions and students will sign up for a 15-minute time slot. During the exam, students will be examined on a randomly chosen topic from the list.
Problem sets will be assigned on Wednesday to be due the following Wednesday on Gradescope. Late problem sets will not be accepted except in the case of an emergency. At the end of the semester, your lowest problem set grade will be dropped from your average. This is meant to accommodate non-emergency absences, so try not to use this unless you have to.
The purpose of the problem sets is for students to practice working with the concepts in the class, solving problems, and explaining complex arguments. Using generative AI to generate solutions does not let you practice these skills, and research shows that it can lead to learning losses:
AI adoption raises homework scores by 18% and reduces completion time by 30%, but lowers monthly exam scores by 20% within six months. High-stakes entrance-exam scores fall by 18 and 24%, with the full penalty emerging only after about two years.
-Strömberg, Lei, Wu, "The Generative AI Learning Penalty: Evidence from Chinese Secondary Education"
If you use an LLM in your problem set, you must acknowledge it at the start of your solution to each problem, for instance "I used [LLM] to [edit/generate] [part/most/all] of this solution". At grader discretion, LLM-generated solutions may receive a flat 95% grade (regardless of correctness) and no further feedback.