I’ve finished my first semester in college and about to start my 2nd one. I’m taking this time to reflect on what I’ve learned, and set new goals for myself.
Exposure to a lot of new subjects has given me a much broader view of mathematics. When I was in high school, I was intensely concerned with Olympiad training, so, in a large part, didn’t know what was beyond high school math.
I studied linear algebra over the summer, and took 18.701 (Abstract Algebra) first semester. One of the things I love about MIT (other than having a lot of friends who like math just as much as I do) is that there is a huge library where I can check out almost any math book I would like to. I read ahead in abstract algebra and group theory, and worked supplementary problems out of Problems in Group Theory, by Dixon.
Anyway, I got to a point where I realized that math in college was way more nonlinear than in high school. I checked out a book on pure group theory, intending to read it front to back, but many of the advanced topics involved subjects beyond group theory, such as category theory and algebraic topology. I realized that I needed to learn a broad range of topics; math is so much more connected. For example, many deep group theory problems require knowledge of representation theory: map the elements to matrices and use the techniques of algebra to solve them. Similarly, algebraic topology applies the techniques of algebra to solve topological problems, and furthermore, the techniques of algebraic topology can be back-applied to solve pure group-theoretic problems! These kinds of relations between different branches of mathematics are everywhere- analytic number theory, algebraic combinatorics,… I’ve never seen category theory before around December, but when I saw it I was amazed at how so much can be derived from something so abstract as to be applicable to completely different parts of mathematics.
The thing is, advanced concepts can sometimes have their proofs “projected” down into simpler areas of math, and still be totally correct and complete. However, this makes it lose a lot of its beauty and intuitive feel- it becomes long and unmotivated. Often textbooks, or teachers, require vital results in teaching their subject that follow more naturally from more advanced topics that may digress too much from their main thrust, giving a sort-of dilemma. However, for me, the more stuff I know, the more things become much easier to understand, even the simple ones.
Some examples spring to mind: I’ve read long, elementary proofs of the Jordan and rational canonical form, but found them to be an extremely slow read. On the other hand, the rcf basically follows from the structure theorem for modules, as does the structure theorem for abelian groups. The orthogonality relations in representation theory can be proved using only basic linear algebra, as Artin does well; though the proof with “inner product= dimension of space of homomorphisms between modules” requires more difficult ideas, is still easier to remember.
During break, I continued with algebra, but also dabbling with category theory and algebraic topology (I say dabble because, as of now, I actually do not know any topology other than a few basic definitions). It’s nice, learning the abstract (as in categoric theory) and “concrete” (though it’s still abstract algebra) concepts side by side, to see how they generalize. As a change of flavor, I also studied graph theory- it has lots of cool problems that don’t require as much material to learn (as say, algebra or analysis) to get where the action is. I learned some more advanced stuff, like the Szemeredi regularity lemma and the nuances of the probabilistic method, but also got to practice recurring problem-solving strategies.
I’ve typed up notes for some of the stuff I’ve learned the past semester (see my old blog), and now I’m ready to move on. This semester I am taking:
- 18.702 Algebra 2
- 18.100C Analysis [Actually there is a schedule conflict as of now, which I hope will be resolved soon.]
- 7.014 Biology
- 21F.406 Modern Chinese Literature and Cinema
I’m going to focus on studying analysis (18.100C is the “expanded” version of the course), and if I have time, I’d like to start topology as well. As I like to do, I’ll get problem books to supplement whatever I’m learning (because doing problems is the best way to learn math!). I’ve also secured a UROP (research project), which I’ll be working on with two other classmates and which I’ll need to learn a lot of algebraic combinatorics for. As for algebra, I’ve looked at almost all the material we’re going to cover, so that I’ll be on the lookout for topics in class that have generalizations I should know and read about (one is that the class covers ideal factorization in complex quadratic rings, but not phrased in terms of Dedekind domains).
There are basically 3 levels of math:
- Reading math textbooks, listening to lectures, etc.
- Doing textbook problems, (2+) Doing harder book problems or contest problems
- Thinking “I wonder if…” and proceeding to try and solve the problem, without really knowing if you’ll get anywhere. Also known as “research.”
My goal is to do more of (2) and (3)! Back in the middle school days, it was all (1), and in the high school days, it was all (2) (or maybe 2.5, since they’re olympiad problems) and no (1) or (3). I did a lot of reading last semester, a lot of (1) and not quite enough (2) to keep up. This semester, I’ll do more of (2), and (3), (3) for my research project, but, not, I hope, just for my research project. (3) is the hardest to do, but it is the crux of math.
(3) might be the hardest but the most natural at the same time. What I mean, which is going to seem more fun to an elementary schooler?
- Reading the textbook.
- Doing textbook problems.
- Playing around with number patterns, or tessellations, or something open-ended.
What got me to thinking about this was that during winter break, I went back to my high school to give a talk at math club. The point was, to get people to come, I needed them to think of math as not just learning stuff from a textbook, but as a method to answer THEIR questions, their curiosity. Rather than presenting an advanced concept, or teaching math competition problems, I decided to take a simple problem and demonstrate how to play around with it to solve it, and how a solution leads to more questions (I talked about repeating decimals- the lengths of periods, and the cycling nature of some of them), which I encouraged the students to explore themselves. [Actually it didn’t go as well as I hoped, but that’s besides the point.]
I’ve been writing this for an hour and it is 2:50am. I think I should go to sleep.