Here is a CV.

My area of expertise is called geometric group theory. Here is an explanation intended for anyone. (By the way, I’d love feedback on this explanation, particularly if you found it confusing or unsatisfying.) In mathematics, a “group” is an algebraic abstraction designed to capture the idea of symmetry. Many things in life and in mathematics have symmetries, and the collection of all symmetries of a thing assemble into what we call a “group”. Group theory, then, is the study of groups. Group theory dates back to at least Évariste Galois in the early 1800s, but of course mathematicians have been interested in symmetry since long before that.

Very often in mathematics, groups are considered part of the “metadata” of another, more central mathematical object. For example, if you have two shapes (mathematicians would say “topological spaces” or “manifolds” or something similar), you might associate one or more groups to these shapes as a way of classifying them: if the groups assigned to each space are different, then you learn from that information that the spaces are different in some way. In this way, fields of study like algebraic topology use groups as a tool to study geometric or topological structures.

Geometric group theory, you could say, flips this script: it posits groups and symmetry as interesting to consider in their own right, and instead seeks to associate to a group some shape or space in order to better reason about the group. As a field, geometric group theory is fairly young, emerging as a discipline in its own right only in the 90s, following groundbreaking work of Mikhael Gromov and William Thurston.

I met geometric group theory as an undergraduate doing a summer research project with Jason Behrstock on properties of a family of groups called right-angled Coxeter groups. While doing that research, I read a bit of Gromov’s article “Asymptotic Invariants of Infinite Groups”. I’d like to quote a bit of the first couple paragraphs of that article for flavor.

I love two things about this passage. For one, I love that Gromov’s geometer is an embodied person, bringing emotions (boredom, joy) and a locational vantage point to their work, rather than being some disembodied mind alone. Second, I love the visual of a collection of discrete points slowly coalescing into a unity, almost the process of zooming out from the atomic to the molecular to the cellular to the human level and beyond. I find that I’m often reaching for a similarly embodied and visual connection with the mathematics that I’m doing, and it’s great when I’m able to reach that kind of understanding.

As applied to mathematics, I share Federico Ardila-Mantilla's axioms:

• Axiom 1. Mathematical talent is distributed equally among different groups, irespective of geographic, demographic and economic boundaries.
• Axiom 2. Everyone can have joyful, meaningful, and empowering mathematical experiences.
• Axiom 3. Mathematics is a powerful, malleable tool that can be shaped and used differently by various communities to serve their needs.
• Axiom 4. Every student deserves to be treated with dignity and respect.

For me, I see mathematics as a science of patterns and analogy. Although most mathematics you and I have encountered may not feel like a persuasive essay, I would posit that mathematics is a form of argument in the way that an essay may be, and is best thought of in that way, both when seeking to understand mathematics, and when seeking to use it to persuade, to prove or to convince. Likewise, although mathematical facts can tend to feel plucked from thin air, it is useful to remember that mathematics (like all the sciences) is part of culture as much as art or any other fundamental human endeavor is. What we know, how we know it, and what we think is worth knowing—mathematical “taste”, is something we are shaping and shaped by constantly.