Employers and Collaborators

About my research, for potential employers and collaborators

I have three primary areas of current research interest.

Computational number theory
Birational geometry and commutative algebra in positive and mixed characteristic
Foundations of analytic geometry, especially p-adic geometry

Within each area of interest, I have much more specialize research expertise. To get an idea of what my actual expertise is, you can take a look at my papers.

I am also actively learning about other exciting trends in these fields, including condensed mathematics and connetions between p-adic geometry and homotopy theory.

I want to highlight a few topics of expertise that might be of general use, even to mathematicians who aren't close enough to my area to want to read my papers.

Firstly, if you find yourself in the following situation:

I have a mathematical algorithm that I'm running in a computer algebra system for my research, and right now it's too slow. I want to know what my options are to speed it up...

then I could be good person to talk to. I'm more or less familiar with what's possible and what's out there, and I've had many similar conversations in the past.

Secondly, if you happen to have an algebraic variety $X$ , and you want to count $\mathbb{F}_{p^n}$ -points (and perhaps deduce the zeta function of $X$ ) via computational methods, but your example is too big for your computer algebra system, then I might be a good person to talk to.

Note that there has been a lot of progress in this area in the last 10-15 years, due to people like David Harvey and Edgar Costa. However, these algorithmic advances have mostly not made their way to widely available computer algebra systems like OSCAR, Magma, and SageMath. One goal of my research is to be a part of fixing this situation, but there's a long way to go.

In the meantime, if you have an example, I am generally aware of most of the algorithms and implementations that are out there, and I can probably tell you whether your example is possible or not.

Finally, related to the previous topic, if you have a variety $X$ that has characteristic $p$ , and you want to calculate the $F$ -crystal structure on the crystalline cohomology of $X$ , then maybe I can help. This is related to the previous item but the $F$ -crystal structure has more information.