Hi! I am a third-year Harvard math graduate student. My advisor is Mike Hopkins.
Before grad school, I did my undergraduate at UT Austin with Dan Freed.

I write both math and physics papers, mostly in between.
My primary research interest is mathematical physics, mostly field theories and anomalies.
Mathematically, I am also interested in knot homologies and homotopy theory.
Besides math and physics, I am also interested in CS (DevOps, kubernetes, type theory and functional programming
languages).

Joint work with Aaron Mazel-Gee,
David Reutter, Catharina Stroppel, and Paul Wedrich. See Catharina's ICM address for motivation and introduction of the ideas, and here is the notes for the address.
Aaron gave a talk here, and here is the slides.

- The Hecke algebras for all symmetric groups taken together form a braided
monoidal category that controls all quantum link invariants of type A and, by
extension, the standard canon of topological quantum field theories in
dimension 3 and 4. Here we provide the first categorification of this Hecke
braided monoidal category, which takes the form of an $\mathbb{E}_2$-monoidal
$(\infty,2)$-category whose hom-$(\infty,1)$-categories are $k$-linear, stable,
idempotent-complete, and equipped with $\mathbb{Z}$-actions. This
categorification is designed to control homotopy-coherent link homology
theories and to-be-constructed topological quantum field theories in dimension
4 and 5.
Our construction is based on chain complexes of Soergel bimodules, with
monoidal structure given by parabolic induction and braiding implemented by
Rouquier complexes, all modelled homotopy-coherently. This is part of a
framework which allows to transfer the toolkit of the categorification
literature into the realm of $\infty$-categories and higher algebra. Along the
way, we develop families of factorization systems for $(\infty,n)$-categories,
enriched $\infty$-categories, and $\infty$-operads, which may be of independent
interest.
As a service aimed at readers less familiar with homotopy-coherent
mathematics, we include a brief introduction to the necessary
$\infty$-categorical technology in the form of an appendix.

Joint work with Salvatore D. Pace. To appear in SciPost.

- In this note, we classify topological solitons of $n$-brane fields, which are
nonlocal fields that describe $n$-dimensional extended objects. We consider a
class of $n$-brane fields that formally define a homomorphism from the $n$-fold
loop space $\Omega^n X_D$ of spacetime $X_D$ to a space $\mathcal{E}_n$.
Examples of such $n$-brane fields are Wilson operators in $n$-form gauge
theories. The solitons are singularities of the $n$-brane field, and we
classify them using the homotopy theory of ${\mathbb{E}_n}$-algebras. We find
that the classification of codimension ${k+1}$ topological solitons with
${k\geq n}$ can be understood understood using homotopy groups of
$\mathcal{E}_n$. In particular, they are classified by
${\pi_{k-n}(\mathcal{E}_n)}$ when ${n>1}$ and by ${\pi_{k-n}(\mathcal{E}_n)}$
modulo a ${\pi_{1-n}(\mathcal{E}_n)}$ action when ${n=0}$ or ${1}$. However,
for ${n>2}$, their classification goes beyond the homotopy groups of
$\mathcal{E}_n$ when ${k< n}$, which we explore through examples. We compare
this classification to $n$-form $\mathcal{E}_n$ gauge theory. We then apply
this classification and consider an ${n}$-form symmetry described by the
abelian group ${G^{(n)}}$ that is spontaneously broken to ${H^{(n)}\subset
G^{(n)}}$, for which the order parameter characterizing this symmetry breaking
pattern is an ${n}$-brane field with target space ${\mathcal{E}_n =
G^{(n)}/H^{(n)}}$. We discuss this classification in the context of many
examples, both with and without 't Hooft anomalies.

Joint work with Arun Debray, Sanath Devalapurkar, Cameron Krulewski, Natalia Pacheco-Tallaj, and Ryan Thorngren.
This is the mathematical companion paper to 2309.16749.

- Smith homomorphisms are maps between bordism groups that change both the dimension and the tangential structure. We give a completely general account of Smith homomorphisms, unifying the many examples in the literature. We provide three definitions of Smith homomorphisms, including as maps of Thom spectra, and show they are equivalent. Using this, we identify the cofiber of the spectrum-level Smith map and extend the Smith homomorphism to a long exact sequence of bordism groups, which is a powerful computation tool. We discuss several examples of this long exact sequence, relating them to known constructions such as Wood’s and Wall’s sequences. Furthermore, taking Anderson duals yields a long exact sequence of invertible field theories, which has a rich physical interpretation. We developed the theory in this paper with applications in mind to symmetry breaking in quantum field theory, which we study in [DDK+24].

Joint work with Arun Debray, Sanath Devalapurkar, Cameron Krulewski, Natalia Pacheco-Tallaj, and Ryan Thorngren. See here for Cameron and my CMSA talk, and here for slides.

- We study defects in symmetry breaking phases, such as domain walls, vortices, and hedgehogs. In particular, we focus on the localized gapless excitations which sometimes occur at the cores of these objects. These are topologically protected by an 't Hooft anomaly. We classify different symmetry breaking phases in terms of the anomalies of these defects, and relate them to the anomaly of the broken symmetry by an anomaly-matching formula. We also derive the obstruction to the existence of a symmetry breaking phase with a local defect. We obtain these results using a long exact sequence of groups of invertible field theories, which we call the "symmetry breaking long exact sequence" (SBLES). The mathematical backbone of the SBLES is studied in a companion paper. Our work further develops the theory of higher Berry phase and its bulk-boundary correspondence, and serves as a new computational tool for classifying symmetry protected topological phases.

Joint work with Arun Debray and Christoph Weis.
Accepted to International Mathematics Research Notices (IMRN). See here for slides.

- We give a geometric construction of the Virasoro central extensions from differential cohomology, using the differential lifts of the
1st Pontryagin differential class. Affirmatively confirming a conjecture of Freed-Hopkins.

Accepted to Comm. Math. Phys..
For an extended version, see my senior thesis. See here for slides.

- We prove a topological version of abelian duality where the gauge groups are finite abelian.
The theories are finite homotopy TFTs, topological analogues of the $p$-form $U(1)$ gauge theories where the gauge group is finite abelian.
Using Brown-Comenetz duality, we extend our results to $\pi$-finite spectra.