Brandon Hanson

About me

I was most recently an assistant professor at the University of Maine in Orono and I am now on the job market. I originally hail from Ottawa, Canada (Go! Sens! Go!).

Outside of math, I like to cook, play guitar and build things out of wood.

Academic timeline

Carleton University

I found my love for math pretty late. In high school, I just about failed calculus. My goal was to be a video game designer, so I decided to major in computer science. But upon enrolling in the requisite first-year course in proofs and discrete math, I learned that math was way more creative than advertised. I switched to a sort of Math/CS double major in my second year and have been hooked ever since.
University of Toronto

Grad school was the first place I started to falter — as an undergrad I was a pretty strong student. At U of T, my classmates were a much stronger bunch and the demands of my courses were pretty high. But through these new challenges, I made great friends and managed to find my way. I had always been interested in number theory. A first-year analysis sequence taught by Larry Guth steered me toward harmonic analysis. So I went into analytic number theory, mentored by John Friedlander. At the same time, Leo Goldmakher was a postdoc at U of T, and he tipped me off to the amazing field of additive combinatorics. This flavour of math was everything I had ever wanted to study, but John wasn’t an expert in the area (though he was wonderfully supportive), so I would have to pick it up on my own. The theorems I proved in my thesis were the result of me trying to understand new ideas or objects in the area.
Penn State University

Leaving grad school was tough. There was an amazing bond with my cohort that came from grinding it out together as students. And State College, PA is a pretty radical change of scene from Toronto. However, Bob Vaughan, my postdoc mentor, made the transition worthwhile. We found some common ground in trying to understand the gaps between sums of squares. At the same time, the ability to work independently helped me come into my own research-wise.

An inverse theorem for the distinct distance problem in Cartesian products was one of the first results I was truly proud of. When I started thinking about the problem I had no idea how to approach it. Being able to see it through was my first indication that I might be able to hack it as a researcher.
University of Georgia

Landing a permanent position in academia was hard, and I was going to have to do another lap of the postdoc track. Fortunately, I had friends in the UGA arithmetic combinatorics group and an amazing wife who was willing to support my academic dream. I spent a lot of time at UGA working with Giorgis Petridis and we proved a pair of results that I am really proud of.
University of Maine

The COVID pandemic didn’t help an already anemic job market. I managed to land a permanent spot at UMaine. Ultimately, my family and I decided to return to Canada following the 2024–2025 academic year. Stay tuned for future updates...
Layer6 AI

After an Erdős Institute bootcamp in data science, I made the move to machine learning. The decade after grad school saw me incorporating more and more probability into my work (even if it wasn’t always apparent), and I came to be very interested in high-dimensional probability. I felt like the ML world was taking off, and I would have more opportunities to make headway on this new direction. It didn’t hurt that the ML world has far more resources than the academic world (or at least resources that I could tap into). As of spring 2025, I have been with Layer6 AI primarily working on reinforcement learning, which is totally rad.

Research

My work focuses on questions that have an equidistribution-theoretic flavour. These tend to lie somewhere in the realm of number theory, combinatorics and harmonic analysis, which combine to form an area of math now referred to as discrete analysis.

The predominant notion that I cling to in my research is the structure versus randomness dichotomy. Basically, this means that a collection of data points either exhibits patterns (structured), or it doesn’t (random). Usually, I’ll aim to prove that data sets with certain properties appear random from some perspective (a sort of equidistribution theorem). Other times, I’ll try to prove that a data set which does not appear random has to be generated by some reasonably simple means (a sort of inverse theorem).

Below are some of the publications which exemplify my research program. For a full list, see my CV.

Select publications

(With M. Rudnev, I. Shkredov and D. Zhelezov) The sum-product problem for integers with few prime factors. Accepted to Compositio. arXiv
(With O. Roche-Newton and S. Senger) Convexity, superquadratic growth, and dot products. Jour. Lond. Math. Soc., 2021. arXiv
(With G. Petridis) A question of Bukh on sums of dilates. Discrete Analysis, 2021. arXiv
(With O. Roche-Newton and M. Rudnev) Higher convexity and iterated sum sets. Combinatorica, 2021. arXiv
Littlewood’s problem for sets with multidimensional structure. IMRN, 2020. arXiv
(With G. Petridis) Refined estimates concerning sumsets contained in the roots of unity. Proc. Lond. Math. Soc., 2020. arXiv
(With O. Roche-Newton and D. Zhelezov) On iterated product sets with shifts II. Algebra and Number Theory, 2020. arXiv
Additive correlation and the inverse problem for the large sieve. Math. Proc. Camb. Phil. Soc., 2018. arXiv
The additive structure of cartesian products spanning few distinct distances. Combinatorica, 2017. arXiv

Teaching

At UMaine, I often teach MAT 228 (Calc 3) and parts of the analysis sequence: MAT 425 and 426 (Real analysis I and II) and MAT 523 and 524 (Graduate real analysis). For the undergrad analysis and grad analysis courses, there are accompanying notes. These notes were inspired by a number of sources, the most prominent being Stein and Shakarchi.

In the summer of 2020, I gave a graduate lecture series on Fourier Analysis and Number Theory, with notes. These notes take from various references including Katznelson’s An Introduction to Harmonic Analysis, Helson’s Harmonic Analysis and Montgomery’s Ten Lectures..., and Tao and Vu’s Additive Combinatorics. These notes may have errors.

About me

Academic timeline

Carleton University

University of Toronto

Penn State University

University of Georgia

University of Maine

Layer6 AI