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Past, Present, and Future Research

Grad school research

Currently, I think about Hopf algebras, modular tensor categories, and generalizations thereof as they pertain to quantum and condensed matter physics. More broadly, my interests lie in an intersection of category theory, algebra, low-dimensional topology, and functional analysis known as quantum topology/quantum algebra. Even more broadly, I like intersections between any seemingly distant ideas in math. For whatever it's worth, I consider myself an algebraist and a topologist. My advisor is Zhenghan Wang.

Publications/preprints in quantum topology/quantum algebra

  1. Modular data of non-semisimple modular categories: Liang Chang, Zhenghan Wang, Qing Zhang, and I wrote a paper on a generalization of modular data to non-semisimple modular categories. The modular data of (semisimple) modular tensor categories are one of their most powerful invariants. In this paper, we studied the notion of modular data from Cohen and Westreich with the goal of seeing what properties of the semisimple case generalize. Moreover, we provide a detailed study in context of the representation theory of two families of examples: $u_q sl_2$ at odd roots of unity and the Drinfeld double of Nichols Hopf algebras at even index, which are a generalization of Sweedler's 4-dimensional Hopf algebra. One particular consequence of this work is a nice disproof of rank-finiteness in the non-semisimple case.
  2. On the formal ribbon extension of a quasitriangular Hopf algebra: I wrote a paper on the ribbon Hopf algebra $\tilde H$ one may construct given a quasitriangular Hopf algebra $H$ by formally adjoining a ribbon element. After completing the previous project, we were left wondering if we can get another example of modular data in the case of odd-index doubled Nichols Hopf algebras. I prove that this algebra is not ribbon. The next question is if we can get an example by formally adjoining a ribbon element, but I prove that the resulting algebra obtained by formally adjoining a ribbon element cannot be modular (in particular, it is not factorizable). The main focus of this article is to understand the representation theory of this formal ribbon extension. I prove that indecomposable objects in $Rep(H)$ double, with simple/projective indecomposable modules in $Rep(H)$ being in one-to-two correspondence with simple/projective indecomposable modules in $Rep(\tilde H)$. I also show that, in the semisimple case, $Rep(\tilde H)$ is isomorphic (not just equivalent) to the sphericalization of $Rep(H)$. Thus, we may consider the formal ribbon extension as a non-semisimple braided version of sphericalization. Finally, I conclude with a case study of odd-index doubled Nichols Hopf algebras. One of the proofs in this article combines the Borel functional calculus and the model completeness of the theory of algebraically closed fields.
  3. Indecomposable Hopf $*$-algebra representations with invariant inner product: As a continuation of our work in the 2024 DRP, Ziqian Zhao and I wrote a note on Hopf $*$-invariant indefinite inner products on topologically indecomposable modules. Hopf symmetry is one of the simplest examples of non-invertible symmetry. In this paper, we studied topologically indecomposable Hopf $*$-algebra modules with invariant inner product and simple submodule, generalizing an important result of Araki. Moreover, we classify invariant Hermitian forms on projective indecomposable $U_q sl_2$-modules at odd roots of unity and indecomposable $H_{n,d}(q)$-modules and investigate our generalization of Araki's theorem in this context.

    4. Active projects

  4. Non-semisimple invariants for lens spaces: In this article with Mitchell Jubeir and Rhea Palak Bakshi, we compute a Kuperberg-type invariant of certain lens spaces and the admissible skein module for all lens spaces as well as handlebodies of arbitrary genus. Very few example computations of these two invariants are known, so our goal was to provide some more nontrivial examples. We do an explicit example using the index-2 Nichols Hopf algebra.
  5. Kernels and cokernels for weak Hopf algebras: In this article, I introduce a notion of kernel and cokernel for maps between weak Hopf algebras (WHAs). This is a nontrivial endeavor because there are no general zero maps in the category of WHAs and WHA homomorphisms. Thus, I define a new type of map defined from relatively separable subalgebras (of which, WHAs with their target subalgebra are an example) to arbitrary algebras, which I call relatively multiplicative maps. I show that certain relatively separable maps, called Hopf maps, from WHAs to Hopf algebras can produce tensor functors. In particular, given a WHA homomorphisms there is a Hopf map which satisfies a modified universal property for cokernels. Moreover, given a Hopf map, there is a WHA homomorphism satisfying a modified universal property for kernels. I investigate when one can get an exact sequence of finite tensor categories from a sequence of WHA homomorphism and Hopf map. Finally, I compute some examples of kernels and cokernels.

Interested in directed reading with me?

If you are interested in directed reading in an official capacity, feel free to contact me in advance; there are lots of topics in math I enjoy. As noted above, I am unofficially an algebraist. However, I have various other interests in topology, analysis, logic, and probability. I would prefer to do readings on topics which aren't offered as courses at UCSB.
My ultimate goal with directed reading is to give sufficient background to work on a research project. I am currently offering undergraduate projects on graphical calculus/fusion categories, ribbon finite tensor categories, bar finite tensor categories, and model theory/functional calculi. Broadly, if you work with me, you'll likely interact with Hopf algebras and fusion categories in some capacity.

Past mentees' papers: Past mentees' posters:

Undergraduate research

Math is cool, and I've done a few cool things with it. In particular, when I was an undergraduate at the Rocheseter Institute of Technology (RIT), I got to try out research in a bunch of different areas of math. RIT was a heavily applied and analysis-focused school, so most of my time there was spent in areas of that general flavor.
My zeroth project was in numerical analysis for general relativity under Manuela Campanelli. If I'm being perfectly honest, I had no idea what was going on. I had never even taken a college physics course! The work was primarily just coding some stuff from other papers. I didn't feel like I was actually doing math. When I realized how unfit I was for that topic, I just found a new, more approachable, more me topic elsewhere. Failure to succeed/enjoy in one research topic is not indicative of how your relationship to research will be as a whole.
My first project was on uncertainty quantification in inverse problems under Akhtar Khan. I started work, like most do, reading some very incomprehensible papers and textbooks. It took me a whole summer to even understand what my research was about. Eventually, I learned that I was working to take a discretization scheme (the stochastic Galerkin method) for a certain direct/forward problem of boundary value problem with random noise $$-\nabla\cdot (\mu\nabla u) = f$$ and make it work for a variety of gradient descent methods for solving the associated inverse problem. Eventually, I had a paper published on this topic.
I got to work on inverse problems some more with Olalekan Babaniyi. In this research, I studied a similar boundary value problem: $$\nabla\cdot (\mu\nabla u) = -\rho\omega^2 u,$$ which has applications to medical imaging. Our work was to generalize/compare five existing approaches to solving this problem. This project has been submitted to a journal and will be linked once it has been accepted.
I had a handful of one-off projects here and there; a couple of which actually produced papers. I worked on a few things with James Marengo and David Farnsworth. One that actually made it to print was a super general result in probability theory:
Theorem. Let $E_1, E_2, \dots, E_n$ be events in an arbitrary probability space. Let $p = \Pr(\bigcap_{i=1}^n E_i)$, and denote by $(i)_n$ the minimal positive integer $\equiv i\pmod{n}$. Then, for any $L \geq 1$, $$\sum_{i=1}^n \Pr(E_i\cap E_{(i+1)_n}\cap\dots \cap E_{(i+L-1)_n})\leq n - L(1-p)$$ with equality holding if and only if $$\sum_{i=1}^n \Pr(E_1\cap E_{2}\cap\dots\cap E_{i-1}\cap E_i^c\cap E_{i+1}\cap\dots\cap E_n) = 1-p.$$
The result actually started with the special case of $L=1$ and $E_i = (X_i < X_{(i+1)_n})$, as we were thinking about intransitive dice, but it blossomed into this.
My other paper with Dr. Farnsworth was never actually published and was just put on the arXiv. It isn't super cutting-edge. We provide a table and a bunch of techniques for converting to and from certain transformations (Legendre-transformations) exactly and approximately. We created this to simply serve as a tool for those using the transformation.
A picture of me giving a presentation on fusion categories to piRIT in undergrad.
A picture of me giving a presentation on fusion categories to $\pi$RIT.
After getting a good feel of numerical analysis and its reliance on functional analysis, I started wondering about other topics that use functional analysis. I searched for topics that used both functional analysis and my not-yet-used love of algebra and stumbled upon operator algebras. After reading a bit about them, I decided on a whim to try to get an REU in the topic, maybe even get a PhD. I got into Ohio State's REU and did work under David Penneys. If we're being perfectly honest, my work was more about fusion categories than operator algebras (though I did get to play with $C^*$-algebras along the way). However, that's what drove me to tensor categories and consequently how I ended up at UCSB. I don't have any papers to show for this, but I do have something equally cool. After working with this research group for the REU, I was brought back to do animation. Over the course of several months, I produced a grand total of, not two, not three, but one video. I really enjoyed myself, and I'm very happy with how the video turned out. Animating the NSF logo at the beginning was one of my biggest accomplishments in life.