Complex variables | Noncommutative Analysis

November 22, 2025

Student projects in complex function theory – The Prime Number Theorem and Riemann’s paper

(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:

On Riemann’s famous paper on the Zeta function – On the Number of Primes Less Than a Given Magnitude Download

3. Other projects

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

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

October 4, 2025

Student projects in complex analysis: The Prime Number Theorem

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). Four their final projects, students were required to choose one of a list of topics on which they wrote a report and gave a lecture. Two of the students, 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.

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January 1, 2021

Seminar talk by Hartz: How can you compute the multiplier norm?

Happy new year!

Next Thursday, January 7th, 2021, Michael Hartz will speak in our Operator Algebras and Operator Theory seminar.

Title: How can you compute the multiplier norm?

Time: 15:30-16:30

Zoom link: Email me.

Abstract:

Multipliers of reproducing kernel Hilbert spaces arise in various contexts in operator theory and complex analysis. A basic example is the Hardy space $H^2$ , whose multiplier algebra is $H^\infty$ , the algebra of bounded holomorphic functions. In particular, the norm of a multiplier on $H^2$ is the pointwise supremum norm.

For general reproducing kernel Hilbert spaces, the multiplier norm can be computed by testing positivity of $n \times n$ matrices analogous to the classical Pick matrix. For $H^2$ , $n=1$ suffices. I will talk about when it suffices to consider matrices of bounded size $n$ . Moreover, I will explain how this problem is related to subhomogeneity of operator algebras.

This is joint work with Alexandru Aleman, John McCarthy and Stefan Richter

October 15, 2020

Three classification results in the theory of weighted hardy spaces in the ball – summary of summer project

Last month we had the Math Research Week here at the Technion, and I promised in a previous post to update if there would be any interesting results (see that post for background on the problems). Well, there are! I am writing this short post just to update as promised on the interesting results.

The two excellent students that worked with us – Danny Ofek and Gilad Sofer – got some nice results. They almost solved to a large extent the main problems mentioned in my earlier post. See this poster for a concise summary of the main results:

FinalPosterDannyGilad Download

Danny and Gilad summarized their results in the following paper. Just take a look. They have some new results that I thought were true, they have some new results that I didn’t guess were true, and they also have some new and simplified proofs for a couple of known results. Their work fits in the long term research project to discover how the structure of Hilbert function spaces and their multiplier algebras encodes the underlying structures, and especially the geometry of sets in the unit disc or the unit ball. More on that soon!

August 24, 2020

Summer project 2020 – Hilbert function spaces of analytic functions in a complex variable

In the week of September 6-11 the Math Department at the Technion will again host the “Math Research Week“, or what we refer to as the the “summer projects week”. As in previous years, I will be offering a project, and this year, with the help of Ran Kiri and Satish Pandey, it will be a project on Hilbert function spaces. See here for the abstract. The purpose of this post is to collect my thoughts and my plans for this project.