Mathematics

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I am interested in homotopy theory, algebraic topology, and algebraic geometry.

My current research explores formal properties of generalized algebraic K-theories. Algebraic K-theory is a subtle invariant of rings and schemes with applications throughout topology, algebraic geometry, and number theory. I am convinced that there is a complex hierarchy of “n-ary” algebraic K-theories, capturing increasingly complicated information, analogous to the chromatic picture in homotopy theory.

I am beginning to be interested in persistent homology and topological data analysis. There are open questions about the foundations of persistent homotopy theory, which are likely related to questions about stratified homotopy theory. I am also interested in that.

Current work

Joint with Aaron Mazel-Gee, I have written a paper giving a universal characterization of the construction taking a scheme $X$ to its stable $\infty$-category of noncommutative motives, $\mathsf{Mot}(X)$. Therafter, we obtain a corepresentability result for secondary algebraic K-theory. You can find the paper on the arXiv at https://arxiv.org/abs/2104.04021. A version with improved formatting is available here: A universal characterization of noncommutative motives and secondary algebraic K-theory.

Secondary algebraic K-theory is in some sense a higher chromatic analogue of ordinary algebraic K-theory. Our results can be seen as analogues of the Blumberg–Gepner–Tabuada theorem on the universal property of algebraic K-theory.

We view this as a fundamental step toward constructing trace maps for secondary algebraic K-theory, and eventually a secondary cyclotomic trace.

In this paper, we also develop some basic theory of presentable enriched $\infty$-categories.

Future work

My project with Aaron lends itself to vast generalizations. I hope to continue this program, which may involve a whole lot more enriched $\infty$-category theory.