Math research

My main math research interests are in noncommutative algebra and algebraic combinatorics, in particular quantized coordinate rings / quantum algebras and their semi-classical limits, and total nonnegativity in real matrices and the real Grassmannian. I’ve also co-authored several papers on applications of total nonnegativity in mathematical physics, specifically Wilson loop diagrams in quantum field theory.

You can find my academic CV here, and a list of publications on the Publications page.

Below I’ve tried to give a very quick lay intro to some of my resesarch areas and potential applications to machine learning. There’s also links to some old math outreach work I did when I was in grad school, and some more technical writings (including my PhD thesis) for anyone interested.

Background

Algebra

In math, algebra is the study of mathematical structures: given a few basic axioms (e.g. existence of addition and multiplication rules, and rules for how they interact), what can we say about the properties of any structure satisfying these axioms? Or, given an example of a structure (say, the set of all 2x2 real matrices, with standard addition and multiplication), what rules can we expect it to follow or not follow?

This is a branch of pure math which can seem a bit abstract and pointless from the outside, but its payoffs are unexpected and unpredictable: usually it ends up being some physicist/biologist/computer scientist hearing about research on a theoretical class of structures and going “Hey, that looks almost like what my atoms/cells/neurons are doing, maybe I can use this?”!

For example, there have been a few really interesting attempts to impose various algebraic structures onto deep learning models; some papers I particularly enjoyed are:

I’d love to see (and to work on!) more research in this direction; the extra structure imposed by requiring a multiplication rule on a vector space could be the link we’re missing in understanding how various machine learning models actually work.

Totally nonnegative matrices

Totally nonnegative (TNN) real matrices are a really cool class of matrix: they’re real matrices with the property that any minor (that is, the determinant of any square submatrix) is nonnegative. One particularly nice thing about this is that they’re much more concrete to explain and to think about than abstract algebraic structures are: you should be able to come up with some 2x2 examples yourself without too much effort!

TNN matrices have some beautiful combinatorics associated with them, as well as some unexpected but very deep links to quantum algebra; this is how I originally got interested in them. They also have multiple real-world applications, especially in areas of physics requiring nonnegative solutions to systems of equations: my favourite such application is in studying a class of solutions to the KP equation, which can be used to model shallow water waves and may lead to new insights in the scince of tsunami prediction.

I’ve yet to see an application of TNN matrices to machine learning (as opposed to nonnegative matrix factorization, which is cool but a completely different technique) but I’m confident it’s out there!

Here are some math outreach things I’ve created or been involved in; they’re aimed at mathematically-inclined people but don’t require any specialist knowledge.

Further reading for mathematicians or math-adjacent people

The following documents go into more detail about my research, and are listed in roughly increasing order of specialisation.