I am a J. L. Doob Research Assistant Professor in the University of Illinois Math Department and a member of the Illinois Quantum Information Science and Technology Center (IQUIST). Recently, I was a visiting professor at UC Santa Barbara, and before that I earned my Ph.D. from UC Davis, where my advisor was Greg Kuperberg. I am partly supported by NSF grant DMS #2038020.

My primary research motivation is to answer questions at the intersection of 3-manifold topology and computational complexity. My work occurs at overlaps of the following subjects:

- (Low-dimensional) geometric topology
- Geometric group theory
- Computational complexity
- Topological quantum field theory (TQFT)
- Quantum computing and quantum information
- Topological condensed matter theory

I identify as a mathematician, not a physicist or computer scientist. My interest is typically in applying ideas from physics and CS to better understand topology and TQFT. (Although occasionally—to Hardy's dismay—the arrow of ideas points in both directions.) For example, I have used ideas inspired by topological quantum computing to prove complexity-theoretic lower bounds for problems in 3-manifold topology.

Here's my CV. Here's a very full, loose lamination with \(\mathbb{Z}/4 * \mathbb{Z}/3\) symmetry:

And here's a video of a talk I gave at the University of Warwick related to my dissertation work: YouTube.

During the fall 2021 semester I am teaching MATH 402 - Non-Euclidean Geometry (Canvas log-in required). In the spring of 2021, I taught MATH 595 - Quantum, Complexity, and Topology. If you want to watch the videos for the latter class you can find them availble for public view at the Illinois Media Space channel here.

My papers and preprints are listed below, along with links to co-authors, any published versions, and arXiv preprints. You might also want to check out my arXiv author page, my MathSciNet author profile (subscription required) or my Google Scholar profile.

**(7)** **Free actions on surfaces that do not extend to arbitrary actions on 3-manifolds.** Accepted for publication in Comptes Rendus - MathÃ©matique. arXiv

**(6)** **Coloring invariants of knots and links are often intractable.** With Greg Kuperberg. Algebraic & Geometric Topology (2021), Volume 21, Issue 3, pp. 1479-1510. arXiv

**(5)** **Haah codes on general three manifolds.** With Kevin Tian and Zhenghan Wang. Annals of Physics (2020), Volume 412, 168014. arXiv

**(4)** **Schur-type invariants of branched G-covers of surfaces.** Topological Phases of Matter and Quantum Computation (2020), *Contemp. Math.*, Volume 747, pp.173-197. arXiv

**(3)** **Computational complexity and 3-manifolds and zombies.** With Greg Kuperberg.
Geometry & Topology (2018), Volume 22, Issue 6, pp. 3623-3670. arXiv

**(2)** **Spaces of invariant circular orders of groups.** With Harry Baik. Groups, Geometry, and Dynamics (2018), Volume 12, Issue 2, pp. 721-763. arXiv

**(1)** **On laminar groups, Tits alternatives, and convergence group actions on \(S^2\).** With Juan Alonso and Harry Baik.
Journal of Group Theory (2019), Volume 22, Issue 3, pp. 359-381. arXiv

My Ph.D. dissertation is titled **Computational Complexity of Enumerative 3-Manifold Invariants** and can be found at the arXiv or ProQuest. It contains the results of items **(3)** and **(6)** above.

E-mail: My last name without any vowels, followed by @illinois.edu

Office: 257C Altgeld Hall

Snail Mail:

Department of Mathematics

1409 West Green Street (MC-382)

Urbana, IL 61801