Welcome to the wonderful world of undergraduate algebraic topology!
Topology, more or less, is the study of spaces, or more specifically spaces where you can reasonably talk about continuity and convergence. Topological spaces can be quite crazy, but we will usually consider fairly nice spaces (like "manifolds" and "CW-complexes"). Some of the most basic questions we can ask are
- What do these things look like?
- How many are there?
- How can we tell two of them apart?
- What interesting properties do they have?
Algebraic topology is a set of tools we can use to try to answer some of these questions. In particular, we will associate algebraic things --- like numbers, vector spaces, polynomials, groups, modules, rings --- to topological spaces. With these algebraic invariants we will be able to do some amazing things. Initially we will consider some questions that seem rather basic, like "how can you tell the difference between the surface of a ball and a doughnut?" and "how can you tell if a closed loop is knotted?", but we will see that in answering these questions we learn surprising things, like "if you want a non-zero vector field tangent to the boundary of a region in three space, say a magnetic field to contain some reaction, then that region better, more or less, look like a doughnut" and "at any given moment there are two antipodal points on the earth that have the same temperature and humidity" and "can certain knots bound a disk like membrane in space-time", and “how can you decompose a group into basic pieces".
Announcements:
- The second midterm exam will cover the material in Section II, through V, that is the material on homework assignments 4 and 5 and part of 6 (though incidental material from earlier in the course might also appear, but I will not be explicitly trying to test that). The best way to prepare for the test is to (1) go through the class notes, (2) go through all the homework problems (even the ones that were not turned in for grading), (3) talk to me if you have questions or read some of the recommended sources. Here is some information about the test:
- The test will be approximately 5 or 6 questions.
- Most will be similar to the homework problems.
- As a "practice test" work problems 1, 5, 6, 9, and 13 from homework 4 and problems 1, 5, 7, and 8 from homework 5, and 1, 6, and 11 from homework 6.
- I guarantee that at least 3 of the problems on the test will either be among these problems or very similar to them.
- One of the questions on the test will consist of several True/False or short answer questions.
- As indicated on the syllabus the second midterm will be April 20.
- The first midterm exam will cover the material covered through the end of Section I, that is the material on homework assignments 1 through 3. The best way to prepare for the test is to (1) go through the class notes, (2) go through all the homework problems (even the ones that were not turned in for grading), (3) talk to me if you have questions or read some of the recommended sources. Here is some information about the test:
- The test will be approximately 5 or 6 questions.
- Most will be similar to the homework problems.
- As a "practice test" work problems 8, 9, and 10 from homework 1 and problems 1, 4, and 5 from homework 2, and 1, 5, and 7 from homework 3.
- I guarantee that at least 2 of the problems on the test will either be among these problems or very similar to them.
- One of the questions on the test will consist of several True/False or short answer questions.
- As indicated on the syllabus the first midterm will be February 23.
- More details on the term paper.
- Guidelines for writing good mathematics by Francis Su
- When you turn in the draft of your paper on March 30rd, 2 other students will give feedback on your paper using this form. (This form is subject to minor modification.)
- By January 31 you must e-mail me with 1) the topic for your term paper, and 2) at least 2 references you will use to explore the topic.
Course Information:
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