Mikhail Khovanov

I can't shake the feeling that Khovanov was a dud. I like the guy, he's sweet, and actually incredibly smart. Look up "Khovanov Homology," he's done a lot of ground breaking research (somehow being used a lot in String Theory). As a Modern Algebra instructor, though, he fell pretty flat. Modern Algebra I is trivial for anyone who's been exposed to a group before. It was weird, somehow it didn't feel like we were ever really doing anything. I think the only interesting stuff was Burnside's Lemma (and group orbits in general), the Sylow theorems and the classification of Abelian Groups. I realized that I was doing fine in the class, but that was just because I had a good idea of how groups worked, not from anything I learned in lecture. The way Khovanov speaks makes it impossible to pay attention. It's not exactly his accent, I can always make out what he's saying at any given point in time, but the way he presents things is just bizarre, and I can't quite explain how. I had no problem paying attention in any other class except for his. Every time I tried to I would lose the thread after 5 - 10 minutes. By the way, this isn't just me: everyone I talked to in the class had the same experience. Modern Algebra II is pretty bizarre. It definitely had a lot more content than Modern Algebra II, going briefly over Ring Theory and then spending a lot of time talking about Fields and Fie…

Professor K is a really awesome guy and I think everyone enjoyed this class. He has a really good sense of humor and is always cracking jokes related to the material. For example, the light-switch started beeping once during class, as if it were going to shut off soon. We had to toggle all three switches to stop the beeping and Michael quipped that it was "the group Z2xZ2xZ2." On another occasion, he remarked "When I was young, I tried to picture the 3-sphere....and I almost succeeded except for one point!" We went through the material in a very non-rushed manner and Michael was happy to answer any question no matter how stupid. The TA (You Qi) had a round table discussion for an hour every week which was great. You Qi is extremely smart and he likes to talk about math. The homework and tests were extremely easy (small computations usually) but the material is rather advanced and there are some difficult patches. There is a wonderful website for the course with many useful links and even a few online books. We loosely followed one of these books for about 2/3 of the course. For the rest of the course Michael came up with this interesting way to present the last topic which wasn't in any book or article but the TA wrote up the notes for this section. They are excellent. What I appreciated most was that we covered necessary topics that you don't see covered in any of the other u…

Topology! Pointset topolgy and basic algebra topology!

I'll just copy and paste the syllabus topics: Rings and commutative rings. Rings of polynomials, residues modulo n and other examples. Matrix rings. Integral domains and fields. Field of fractions. Homomorphisms of rings and ideals. Quotient rings and First Isomorphism Theorem for rings. Principal ideal domains and polynomial rings over fields. Prime and maximal ideals. Irreducible polynomials. Euclidean domains and unique factorization domains. Characteristic of a field. Finite fields. Linear algebra over a field. Field extensions and splitting fields. Field extensions of field automorphisms. Galois group. Solvability by Radicals. Ruler and compass constructions. Independence of characters. Galois' Theorems. Applications. Fundamental Theorem of Algebra. Applications of finite fields.

25% weekly homework, 10% quizzes (2 or 3), 15% each midterm, 35% final.

1 problem set every week,2 tests, 2 quizzes (All of them very straightforward, no tricks)

There were essentially two "parts" to the course—in the first half, we covered many of the basic concepts of general topology (continuity, compactness, connectedness), but didn't talk in depth about separation or countability axioms, because Prof. Khovanov was interested in delving into algebraic topology as soon as possible. We then learnt about the fundamental group, the lifting property of covering spaces, and the van Kampen theorem.

I learned the fundamental concepts of Field, Ring, and Galois theory, which help to understand the general tools one can use to manipulate and reduce polynomials under different fields.

The first half of the course covered an introduction to point-set topology, with some special topics such as the space-filling curve (following Munkres Ch. 2 and 3), and the second half covered basic algebraic topology (drawing on Munkres, roughly sections 51 onwards, with some use of Hatcher and other sources to supplement). We ended with the van Kampen theorem.

We began with ring theory, and built up to Galois theory by the end of the semester (which then brought together this course's focus with the content covered last semester in Modern Algebra I).