Hanson

334 Neville Hall, University of Maine

brandon.hanson1@maine.edu

I am currently an assistant professor at the University of Maine in Orono, but I originally hail from Ottawa, Canada (Go! Sens! Go!).

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

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.

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 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.

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.

Landing an 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.

The COVID pandemic didn't help an already anemic job market. Fortunately, I managed to land a permanent spot at UMaine. Our department is small, but the people in it make excellent colleagues.

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**.

- (With M. Rudnev, I. Shkredov and D. Zhelezov)
**The sum-product problem for integers with few prime factors**.*Submitted*. 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

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 (Grad. 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 Addtive Combinatorics. These notes may have errors.