Noncommutative Analysis

(This post is an updated version of an older post with another project added)

In the recent spring semester I taught the advanced course Function Theory 2, which was about a number of advanced topics in complex function theory, where “advanced” means that they are typically not covered in a first course in complex function theory (here is the info page for the course to get an idea of what is was about). For their final projects, students were required to choose one of a list of topics on which they wrote a report and gave a lecture. Some of the students agreed to share their projects online, and I am putting them up here for posterity 🙂

1. The Prime Number Theorem

Gal Goren and Yarden Sharoni asked me if they could prepare a video instead of a lecture. Even though I estimated that this would be about ten times more difficult than giving a talk, I agreed. I am very happy to share their final project here, which was beautifully done.

Perhaps it is worth saying that the video is not entirely self contained, and it does require the viewer to know stuff about holomorphic functions, meromorphic function, and specifically to know quite a lot about the Riemann zeta function. All the prerequisites for this video were taught in the lectures, however, I left the Prime Number Theorem and the most challenging facts needed about the Riemann zeta function to the project. However, in the video Gal and Yarden explain all the facts that they use, and a viewer with standard undergraduate complex analysis background that is willing to take that on faith some facts will be able to enjoy this video (Gal and Yarden are planning to prepare another video which will contain all the prerequisites, so I subscribed to their channel looking for to that).

2. What did Riemann do in his famous paper?

Uri Ronen and Tom Waknine chose one of the most challenging topics offered: to read Riemann’s paper “On the number of primes less than a given magnitude” write a report on it and give it to a lecture to the class, explaining what this paper achieves. I suggested to use Edwards’s book “Riemann’s Zeta Function” which contains a translation of the paper and begins with a chapter walking the readers through the paper. The excellent reference notwithstanding, this was a very challenging project and Uri and Tom gave a masterful lecture. Here is Uri and Tom’s report:

3. Other projects

Other projects by the students (which I will not upload) were on:

1. The Beurling-Lax-Halmos Theorem on invariant subspaces of the shift.
2. Basic theory of Dirichlet series.
3. Rudimentary theory of elliptic functions.
4. The Paley-Wiener Theorems.
5. Caratheodory’s interpolation theorem.