Of course, this is a rant. You can tell by the title. But, for the record, let me be forward that I don’t have an idea how to solve the problem that I will be lamenting, nor do I think that I am doing a better job than everyone else at assessing mathematics done by other people. I am writing this little rant as the beginning of a thought process about how to improve things. Now that this little apology is out of the way, let me lament away.
Once in a while I compare the rigor and care that I exercise when checking whether a piece of mathematics is correct, with the methodology that I employ when evaluating the quality of mathematical output, such as when I referee a paper for a journal, or write a report for a grant funding agency, when considering job applicants and even when judging the worth of my own work. The difference is like earth and sky. I believe that I am not alone in this.
This document contains some excerpts from Part B2 of my ERC grant proposal. The area of NC function theory is not as widely recognized as some other areas competing for grants, I therefore thought that it would be interesting for some readers if I told the mathematical story of how I was led to enter this area. My proposal ended up not being funded, and I thought that it might be of use to somebody out there if I made the expository parts of my proposal available online.
In a recent academic assembly a colleague said something along the lines of “the AI revolution is imminent, we must act quickly or we will be left behind” which – together with the incoherent clamor coming from all directions (ministry of education, university administration, colleagues, students, social media) – has led me to realize the extent of the confusion in which higher education finds itself today.
There is no new thing under the sun. This is not the first time that we’ve seen the academic community under the spell of a collective urge to scramble and catch the future by its tail. But the current anticipation of the rise of AI seems different: this time it’s justified, this time it’s a true revolution. Okay. So before stepping out to face the storm, let’s take a deep breath and get our thoughts in order.
(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:
Malte Gerhold, Marcel Scherer and I have recently posted our paper “Empirical bounds for dilations of free unitaries and the universal commuting dilation constant” to the arxiv. This my first paper that is in experimental mathematics. What we do in it is gather evidence for the conjecture that the universal commuting dilation constant is strictly less than . The universal commuting dilation constant is the minimal constant such that for every pair of contractions on a Hilbert space , there exists a pair of commuting normals on a larger Hilbert space such that that dilates , that is, such that
is strictly less than 
. The universal commuting dilation constant 
such that for every pair of contractions 
on a Hilbert space 
, there exists a pair of commuting normals 
on a larger Hilbert space 
such that 
that dilates 
Noncommutative Analysis 