January 09, 2024.

DRP Türkiye

The Directed Reading Program Türkiye (in their words) is an online program that pairs undergraduate students studying mathematics at universities in and around Türkiye with graduate students and young researchers at institutions around the world to work together on selected books or articles in mathematics during the summer. I was one of the mentors when it first started in 2021 and I joined again as a mentor in 2023.

My motivation was two-fold: I wanted to give back to the Turkish mathematical community by pairing up with a student from Türkiye and I wanted to learn more about the recent work on Arf local rings. So, when I filled the application form for mentors, I chose commutative algebra and representation theory.

But DRP Türkiye had grown up to become an international thing in 3 years and representation theory is a broad subject area. Hence, I ended up with a mentee from India who is interested in harmonic analysis!


After an initial meeting with Gargi (Biswas), I decided to forget about Arf rings for this project. After spending some google-time, we have found a paper that would be of interest to both of us: Abstract harmonic analysis, homological algebra, and operator spaces. (V. Runde, Contemp. Math. 328, 263--274 (2003)). So, Gargi taught me some harmonic analysis and I spoke about homological algebra when we met and it was a fun little project.

As Gargi writes in her report, discussions on amenability were essentially started in 1904 by Lebesgue from a measure theoretic view. Then, in the 1930s, von Neumann introduced the original definition of amenable locally compact groups in his study of Banach-Tarski paradox. The term amenable was given by May in 1950 as a pun since (one of the equivalent definitions of amenability says that) we define amenability via invariant means. The Wikipedia page on amenable groups lists several equivalent conditions for amenability.

The main theorem we were interested in was due to B.E. Johnson who proved in 1972 that a locally compact group $G$ is amenable if and only if certain Hochschild cohomology groups of its convolution algebra $L^1(G)$ vanish. In very broad terms, this theorem states that amenability (which is a property that appears when one wants to do calculus on groups) can be checked by homological algebra (which is essentially advanced linear algebra).

If you have 20 minutes to spare, Gargi talked about this reading project in the DRP Türkiye Symposium and below is her presentation.

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