Prof Alex Bartel

Published: 9 February 2022

Algebraic Number Theory

Alex Bartel Photo

Prof Alex Bartel
School of Mathematics & Statistics

EPSRC Early Career Fellowship
2017 – 2022

Area of Research

Algebraic Number Theory


Why did you choose to pursue a fellowship in your research career?

I had the idea that a widely believed conjecture in number theory might actually be false, and had a concrete strategy in mind for how to disprove it. This idea also pointed to the fact that the conjecture was formulated, fundamentally, in the wrong terms, describing, as it were, an emergent phenomenon rather than the primary one. I therefore decided to ask for a very substantial part of my own time and for a postdoc in order to develop this idea into a full-fledged research programme.

Why work at the University of Glasgow?

When I joined Glasgow, I became the only number theorist here. However, Glasgow has very strong and diverse groups in algebra and geometry, areas of great interest to me, and I had already had the pleasure of interacting with the mathematicians in those neighbouring areas. I very much liked the atmosphere in the department, the interdisciplinary and inclusive ethos of the Pure Mathematics group, and I was excited by the prospect of building up a number theory group in such an environment.

How would you describe your research in 20 words or less?

I study number theoretic objects not by investigating each one under a magnifying glass, but instead zooming out and squinting.

What is your research highlight?

The Cohen—Lenstra heuristics are a central web of conjectures in the area of Arithmetic Statistics, and several Fields Medals in recent years were awarded for proofs of special cases. In joint work with Hendrik Lenstra, we have disproved the Cohen—Lenstra conjectures, and have set out a programme for how to fix and generalise them. We have also successfully carried out a large part of this programme, thus making the conjectures applicable to many more objects that did not satisfy the basic hypotheses of the original heuristics; but the remaining work is substantial and fascinating.

What do you look for in a collaboration?

I like to work with people with whom we can bounce ideas off each other efficiently. In my best collaborations, it often happens that A echoes back to B what A thinks B just said, but B is surprised at the new insight. Thus, every idea gets fermented a bit more in one brain, and then gets transferred to the other to continue fermenting. I have had the great fortune of making surprising discoveries this way, where afterwards both people would credit the other with the main idea.

How do you see your research impacting society?

Historically, it has often taken decades or even centuries for research in number theory to directly impact society, but when it does, it often does so in huge ways. For example modern secure communication, internet, and banking would be impossible without number theory, but the theory used was often developed decades earlier, and without a view to these applications. A more immediate and more direct impact will come from my engagement with students of all ages, be it via outreach or at university level. It might be hard to draw direct links from my research to outreach activities, but my research certainly influences how I think about mathematics, and therefore also how I talk about it to others.

What next?

My fellowship is coming to an end in 2022. It has achieved its stated goals, but in the process it has flung open many new doors, which need to be explored, which I am looking forward to. I am also looking forward to continuing to build a flourishing number theory group at Glasgow: when I joined in 2017, I became the only number theorist in Scotland. Now we have 2 staff, 3 postdocs, 3 Ph.D. students, and between us 3 EPSRC grants and 2 Carnegie Fellowships. The future for Glasgow number theory looks bright.
On the other hand, because of my buy-out I have not been too involved in teaching for the past 5 years, and in the short-term future I am looking forward to getting more direct contact with students.

First published: 9 February 2022