Analytic Number Theory

May 2015

Dorian Goldfeld is a great guy! He's one of Columbia's most accomplished math professors, and that definitely bleeds through into the class. Often he'll prove things in non-standard ways that's he's encountered over the course of his career that are a little nicer than the regular way. One time he showed us a proof for a theorem that he made (and published) by himself, which was far shorter than the regular one. Having said this, some technical stuff will go over your heard. I wouldn't say he races through the material, but he's definitely not slow. This man is so comfortable with the material that he does pretty involved integrals and change-of-variables in a single step. The first part of the course goes over the historical proof of the prime number theorem, and in the process covers an extraordinary amount of mathematical machinery, filled to the brim with incredible tricks I wouldn't have thought of in a million years. I wish at the end he gave a birds-eye view of the theorem we spent so long proving, so I could remember exactly how it all fit together. I think it got pretty myopic at points, but I can't say I didn't learn a lot in the process. After the first part of the course I was a litte disillusioned because I felt that I knew a bunch of scattered techniques without feeling I understood the subject from the ground-up. This was remedied in the second part of the course, which was primarily focused on modular forms. I gotta admit, I really liked those modular forms. There was definitely an understandable component to them, even if the results got a little crazy at times. The problem sets were actually pretty good. They were all 4 problems long and took over a day or two to complete, but not too much more. I had a really hard time with the first one because I had no idea what analysis was, but after that I was able to attack them with confidence. You'll DEFINITELY need to take good notes in able to do these problems, as they often require a result proved in class. The last problem set was the best, you got to prove a few results that looked like complete black-magic, and all on your own! The final was reasonable. It too used things proved in class (and how they were proved). It was a take-home final over the weekend that lasted 24 hours. The class was very small, all filled with bright kids who understood mostly everything, and they all got good grades from what I hear. Goldfeld isn't a grade monster. The only prereq for the class is complex variables, but if you know all about the residue theorem and some facts about holomorphic / meromorphic functions, you'll be okay. Furthermore, I think a general mathematical interest is good to have, as he uses some group theory and linear algebra that aren't technically pre-reqs. Having said all this, the class is mostly self-contained. Also, a word must be said for Goldfeld the man. He laughs a lot while teaching, and is generally very friendly and lighthearted. He talked to me on a few occasions, and genuinely seemed invested in having his students understanding the material. Sometimes he tells great anecdotes about himself or other mathematicians that bring a great character to the material. I would definitely recommend this class, or any upper-level class taught by Goldfeld. I think this was a relatively painless and fascinating introduction to Analytic Number Theory, which has become a vast branch of math. You learn a lot of landmark results, and break a lot of ground into the subject. I can read wikipedia articles about the subject matter and understand almost all of it. 10/10