The whole theory has been, to a great extent, inspired by ideas that arose in theoretical physics. … The development of this subject shows once more that physics and mathematics intercommunicate and influence each other. — Vladimir G. Turaev
My research is in low-dimensional topology, with an emphasis on 3-manifolds and links. My thesis work lies in quantum topology, a field that originated with the discovery of the Jones polynomial and ideas from theoretical physics. It is built around quantum invariants and structures known as topological quantum field theories (TQFTs).
The central aim of my thesis is to understand how quantum invariants reflect the geometry of a 3-manifold. Here the geometry comes from Thurston’s geometrization theorem, under which every 3-manifold can be cut along spheres and tori into geometric pieces. On the quantum side, I study the Turaev–Viro invariants, introduced through a state-sum model on a triangulation of a 3-manifold. The generalized Turaev–Viro volume conjecture makes this connection precise: it predicts that the large-level growth rate of these invariants recovers the simplicial volume of the manifold. This is the 3-manifold analogue of the Kashaev and Murakami–Murakami volume conjecture for knots; it was formulated by Chen and Yang for hyperbolic manifolds and extended to all 3-manifolds by Detcherry and Kalfagianni.
My thesis establishes the conjecture for several large classes of 3-manifolds, using three different sets of tools: analytic estimates, TQFT methods, and Ohtsuki’s saddle-point method, supported by computer experiments in Python.
Let $M$ be a compact oriented 3-manifold with empty or toroidal boundary, let $r \geq 3$ be odd, and let $TV_r(M, q)$ denote the $SO(3)$ Turaev–Viro invariants. The simplicial volume $\mathrm{Vol}(M)$ is the sum of the hyperbolic volumes of the hyperbolic pieces in the JSJ decomposition of $M$. Set
$$LTV(M) := \limsup_{r \to \infty,\ r \text{ odd}} \frac{2\pi}{r} \log \left| TV_r\!\left(M, e^{2\pi i / r}\right) \right|.$$Conjecture (Generalized Turaev–Viro Volume Conjecture). For every compact orientable 3-manifold $M$ with empty or toroidal boundary, $LTV(M) = \mathrm{Vol}(M)$.
Single-author paper. arXiv:2504.10682 · Submitted to the International Journal of Mathematics
Using analytic tools, I proved the volume conjecture for large families of oriented Seifert fibered 3-manifolds with empty or non-empty boundary, by studying the $SO(3)$ Turaev–Viro invariants at $q = e^{2\pi i/r}$ and their asymptotic behavior as $r \to \infty$. Starting from Hansen’s explicit formula for the Witten–Reshetikhin–Turaev invariants of the double $D(M)$, I found a condition on the Seifert invariants under which almost all terms of the invariants vanish. This makes the computations tractable and gives evidence for why the conjecture is formulated using a $\limsup$ rather than a $\lim$: the invariants vanish for infinitely many $r$, so the growth rate is attained only along a subsequence.
The result covers, for example, Seifert fibered manifolds whose exceptional fibers have pairwise coprime multiplicities, as well as complements of links with zero simplicial volume in these manifolds.
Joint with Renaud Detcherry and Efstratia Kalfagianni. arXiv:2505.01546 · Submitted to the Journal of the London Mathematical Society
Most results in this area rely on direct analytic estimates. Since the Witten–Reshetikhin–Turaev invariants carry the structure of a TQFT, it is natural to ask whether that structure, together with 3-manifold topology, can replace brute-force analysis. We studied how the conjecture behaves when a Seifert fibered manifold $S$ is glued to a manifold $M$ with toroidal boundary. Decomposing $S$ into elementary cobordisms and bounding the operator norms of the associated TQFT maps, we showed that the volume conjecture is closed under this gluing operation. As applications, the conjecture holds for:
Joint with Ka Ho Wong and Efstratia Kalfagianni.
Every extension of Ohtsuki’s method so far concerns surgeries or fillings on a single knot complement, so manifolds obtained by gluing two hyperbolic knot complements are a natural next step. Let $M_p$ be the closed 3-manifold obtained by gluing two figure-eight knot complements along their boundary tori by $p$ positive Dehn twists, so that $\mathrm{Vol}(M_p) = 2\,\mathrm{Vol}(4_1) \approx 4.0598$. Using Ohtsuki’s method (Poisson summation and multivariable saddle-point analysis), we obtained the asymptotic expansion of the invariants and proved the volume conjecture for $M_p$ for every $p \in \mathbb{Z}$.
A Python implementation I developed, which computes the growth rate for $r$ up to 20,000 in under a minute, confirms the result. It also gives evidence for the figure-eight/trefoil gluing, where one piece is Seifert fibered: there, the growth rate comes only from the hyperbolic piece.
| $p$ | −3 | −2 | −1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|---|
| $4_1 \cup 4_1$ | 4.0618976 | 4.0619119 | 4.0619212 | 4.0619244 | 4.0619212 | 4.0619118 | 4.0618973 |
| $4_1 \cup 3_1$ | 2.0297096 | 2.0297112 | 2.0283336 | 2.0297131 | 2.0283002 | 2.0297125 | 2.0297122 |
Computed growth rates at $m = 10{,}000$ (where $r = 2m+1$). Targets: $2\,\mathrm{Vol}(4_1) = 4.0597664$ (top row) and $\mathrm{Vol}(4_1) = 2.0298832$ (bottom row).
Closed Seifert fibered manifolds and lens spaces. For general closed Seifert fibered 3-manifolds, including lens spaces and small Seifert fibered spaces, the conjecture remains open. I plan to develop an approach that simplifies the invariants directly, without passing to the double, starting with small Seifert fibered manifolds with three exceptional fibers.
Hyperbolic cobordisms. Can the TQFT operator methods from Project 2 be pushed to gluings along hyperbolic pieces, which carry volume and must add to the growth rate? Can the argument be extended to Seifert pieces with a single boundary component?
Beyond the figure-eight knot. Can Ohtsuki’s method be adapted to prove the mixed figure-eight/trefoil case, and can the figure-eight knot be replaced by other hyperbolic knots, such as the $5_2$ knot?
Computation and software. I plan to release my code as a documented open-source package for Reshetikhin–Turaev and Turaev–Viro invariants of gluings at large $r$, and to use it for systematic experiments where no theorem exists yet. This work is well suited to undergraduate and beginning graduate projects: the invariants are elementary to define, and the results can be checked immediately against geometry.
Quantum computing. Quantum invariants provide important mathematical structures for quantum computing: topological objects can represent quantum states, and operations such as braiding can represent quantum computations. In the longer term, I aim to use quantum invariants to study topological quantum computers, particularly systems based on anyons and braiding, and, conversely, to explore how quantum-computing techniques can provide new tools for computing difficult quantum invariants.