This is the homepage of Alex Cowan's summer 2025 Directed Research Program (DRP) for underrepresented demographics, organized by the Women in Mathematics (WiM) committee at the University of Waterloo.
DRP coordinators:
Kateryna Tatarko ktatarko@uwaterloo.ca
Sophie Spirkl sspirkl@uwaterloo.ca
WiM general inquiry: wim@uwaterloo.ca
Many mysterious sequences of arithmetic interest, such as the number of divisors of integers or the number of solutions to polynomial equations over finite fields, magically arise in the Fourier coefficients of certain very structured analytic objects called automorphic forms. This project is about answering questions in arithmetic statistics by extracting information from automorphic forms using methods from analytic number theory.
Learning these methods unguided can be daunting, as papers in the field are often long, technical, and dense with notation. This project's guided research will impart practical working knowledge of the rather intuitive underlying techniques, so that one can both more easily understand the literature as well as calculate for one's self, which is quite satisfying and maybe even a little fun.
This project will teach students to use "spectral methods for GL2 automorphic forms" to compute "shifted convolutions". These techniques are fundamental in a large subfield of analytic number theory. I taught a topics course in 2023 on this subject, and have a YouTube playlist of lecture notes.
The first problem we'll tackle is computing correlations of certain divisor sums by applying "spectral methods" to "holomorphic Eisenstein series". This problem also involves the use of a lesser-known technique sometimes called "automorphic regularization". This problem can definitely lead to a paper.
If we complete this problem with time to spare, the DRP can continue in several ways, as the techniques it teaches are very general and flexible, and constitute a sizable chunk of an entire perspective on or approach to number theory. It's quite plausible students may be inspired to find their own applications. One of several possible continuations to the project would be to improve a paper of mine which applies "spectral methods" to "real analytic Eisenstein series" by making the error term effective in various "aspects" I neglected for simplicity the first time around.
When doing research, it is normal, in fact inevitable, to have gaps in background knowledge. An important skill for research mathematicians is being able to pick things up as needed, and this DRP is a good an opportunity as any to practice that skill. I encourage every student to apply, and I'm more than happy to help fill in any gaps.
When I taught the 2023 course I said that, while it was advanced in that it aims to get students to research-level proficiency, there were basically no prerequisites. In particular the syllabus read "Familiarity with a minimal amount of basic complex analysis, e.g. Cauchy's residue theorem, helps, but can be picked up on the fly." The first video in the YouTube playlist covers pretty much all you need to know. However, if you've done complex analysis and found that it wasn't your cup of tea, then this project may not be the best for you. There is a lot of complex analysis.
This page was last updated 2025/03/30.