Campbell Wheeler

I have been a PhD student at the Max Planck Institute for Mathematics since September 2018. My supervisor is Don Zagier and am co-supervised Stavros Garoufalidis. Before beginning my PhD I completed my masters degree at the university of Melbourne where I was supervised by Paul Norbury.

## Interests

Maths: I'm interested in interactions between geometry, low dimensional topology, number theory, and physics. In particular I like computing quantities, arising from enumerative geometry and quantum topology, and exploring their geometric, combinatorial and number theoretic properties.

Nonmaths: I enjoy playing music and play the saxophone well-ish and attempt to play other things. If you want, you can check out my old band from Melbourne The Cactus Channel. Living in Bonn and doing mathematics means that I of course boulder and I'm also always up for games of any description.

## Thesis

My PhD project is currently focused on investigating the asymptotic and quantum modular properties of q-hyper-geometric functions. These functions are important in quantum topology where they arise as invariants of three manifolds. q-holonomic modules and state integrals provide some of the most important tools in their study. q-hypergeometric functions also provide an interesting playground to study resurgence both at the classical and q level.

## CV

You can find my CV here.

## Papers

The 3D-index of Dimofte--Gaiotto--Gukov is an interesting collection of $q$-series with integer coefficients parametrised by a pair of integers and associated to a 3-manifold with torus boundary. In this note, we explain the structure of the asymptotic expansions of the 3D-index when $q=e^{2\pi i\tau}$ and $\tau$ tends to zero (to all orders and with exponentially small terms included), and discover two phenomena: (a) when $\tau$ tends to zero on a ray near the positive real axis, the horizontal asymptotics of the meromorphic 3D-index match to all orders with the asymptotics of the Turaev--Viro invariant of a knot, in particular explaining the Volume Conjecture of Chen--Yang from first principles, (b) when $\tau \to 0$ on the positive imaginary axis, the vertical asymptotics of the 3D-index involves periods of a plane curve (the $A$-polynomial), as opposed to algebraic numbers, explaining some predictions of Hodgson--Kricker--Siejakowski and leading to conjectural identities between periods of the $A$-polynomial of a knot and integrals of the Euler beta-function. This is joint work with Stavros Garoufalidis.

We introduce the notion of modular q-holonomic modules whose fundamental matrices define a cocycle with improved analyticity properties and show that the generalised q-hypergeometric equation, as well as three key q-holonomic modules of complex Chern--Simons theory are modular. This notion explains conceptually recent structural properties of quantum invariants of knots and 3-manifolds, and of exact and perturbative Chern--Simons theory, and in addition provides an effective method to solve the corresponding linear q-difference equations. An alternative title of our paper, emphasising the equations rather than the modules, is: Modular linear q-difference equations. This is joint work with Stavros Garoufalidis.

Some years ago, it was conjectured by Garoufalidis that the Chern-Simons perturbation theory of a 3-manifold at the trivial flat connection is a resurgent power series. We describe completely the resurgent structure of the above series (including the location of the singularities and their Stokes constants) in the case of a hyperbolic knot complement in terms of an extended square matrix of $(x,q)$-series whose rows are indexed by the boundary parabolic $\mathrm{SL}_{2}(\mathbb{C})$-flat connections, including the trivial one. We use our extended matrix to describe the Stokes constants of the above series, to define explicitly their Borel transform and to identify it with state-integrals. Along the way, we use our matrix to give an analytic extension of the Kashaev invariant and of the colored Jones polynomial and to complete the matrix valued holomorphic quantum modular forms as well as to give an exact version of the refined quantum modularity conjecture of Garoufalidis and Zagier. Finally, our matrix provides an extension of the 3D-index in a sector of the trivial flat connection. We illustrate our definitions, theorems, numerical calculations and conjectures with the two simplest hyperbolic knots. This is joint work with Stavros Garoufalidis, Jie Gu and Marcos Mariño.

The volume $\mathcal{B}_{\Sigma}^{\mathrm{comb}}(\mathbb{G})$ of the unit ball -- with respect to the combinatorial length function $\ell_{\mathbb{G}}$ -- of the space of measured foliations on a stable bordered surface $\Sigma$ appears as the prefactor of the polynomial growth of the number of multicurves on $\Sigma$. We find the range of $s\in\mathbb{R}$ for which $(\mathcal{B}_{\Sigma}^{\mathrm{comb}})^s$, as a function over the combinatorial moduli spaces, is integrable with respect to the Kontsevich measure. The results depends on the topology of $\Sigma$, in contrast with the situation for hyperbolic surfaces where Arana-Herrera and Athreya (arXiv:1907.06287) recently proved an optimal square-integrability. This is joint work with Gäetan Borot, Séverin Charbonnier, Vincent Delecroix, Alessandro Giacchetto.

We study the combinatorial Teichm ller space and construct on it global coordinates, analogous to the Fenchel-Nielsen coordinates on the ordinary Teichm ller space. We prove that these coordinates form an atlas with piecewise linear transition functions, and constitute global Darboux coordinates for the Kontsevich symplectic structure on top-dimensional cells. We then set up the geometric recursion in the sense of Andersen-Borot-Orantin adapted to the combinatorial setting, which naturally produces mapping class group invariant functions on the combinatorial Teichmuller spaces. We establish a combinatorial analogue of the Mirzakhani-McShane identity fitting this framework. As applications, we obtain geometric proofs of Witten conjecture/Kontsevich theorem (Virasoro constraints for $\psi$-classes intersections) and of Norbury's topological recursion for the lattice point count in the combinatorial moduli spaces. These proofs arise now as part of a unified theory and proceed in perfect parallel to Mirzakhani's proof of topological recursion for the Weil-Petersson volumes. We move on to the study of the spine construction and the associated rescaling flow on the Teichm ller space. We strengthen former results of Mondello and Do on the convergence of this flow. In particular, we prove convergence of hyperbolic Fenchel-Nielsen coordinates to the combinatorial ones with some uniformity. This allows us to effectively carry natural constructions on the Teichm ller space to their analogues in the combinatorial spaces. For instance, we obtain the piecewise linear structure on the combinatorial Teichm ller space as the limit of the smooth structure on the Teichmuller space. To conclude, we provide further applications to the enumerative geometry of multicurves, Masur-Veech volumes and measured foliations in the combinatorial setting. This is joint work with Jørgen Andersen, Gäetan Borot, Séverin Charbonnier, Alessandro Giacchetto, Danilo Lewanski.

We study Masur-Veech volumes $MV_{g,n}$ of the principal stratum of the moduli space of quadratic differentials of unit area on curves of genus $g$ with $n$ punctures. We show that the volumes $MV_{g,n}$ are the constant terms of a family of polynomials in n variables governed by the topological recursion/Virasoro constraints. This is equivalent to a formula giving these polynomials as a sum over stable graphs, and retrieves a result of [DGZZ] proved by combinatorial arguments. Our method is different: it relies on the geometric recursion and its application to statistics of hyperbolic lengths of multicurves developed in [ABO]. We also obtain an expression of the area Siegel-Veech constants in terms of hyperbolic geometry. The topological recursion allows numerical computations of Masur-Veech volumes, and thus of area Siegel-Veech constants, for low $g$ and $n$, which leads us to propose conjectural formulas for low $g$ but all $n$. We also relate our polynomials to the asymptotic counting of square-tiled surfaces with large boundaries. This is joint work with Jørgen Andersen, Gäetan Borot, Séverin Charbonnier, Vincent Delecroix, Alessandro Giacchetto, Danilo Lewanski.

## Collaborators

People I have worked with since starting the PhD are:

## Masters Thesis

You can find my old masters thesis here.

Last modified: August 30th 2022.