Campbell Wheeler

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 by Gaëtan Borot and 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 number theory, low dimensional topology, geometry 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-hypergeometric functions. These functions are important in quantum topology where they often arise as invariants either as q-series or elements of the Habiro ring. One application is providing a precise quantum modularity conjecture for WRT invariants of some closed 3-manifolds and its proof in some examples. Moreover, I wish to compare this to formal gaussian integration and surgery calculations along with recent developments of q-series invariants for 3-manifolds, resurgence and state integrals. You can find my old masters thesis here.

Preprints

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.

• Topological recursion for Masur-Veech volumes.

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 Teichmüller 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 Teichm ller 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.

• On the Konsevich geometry of the combinatorial Teichmüller space.

We prove that, the function which associates the Thurston volume of the unit ball with respect to the combinatorial length, is integrable over the combinatorial moduli space. Moreover, we describe how the function can in fact be computed explictly in terms of edge lengths in each cell. This is joint work with Gäetan Borot, Séverin Charbonnier, Vincent Delecroix, Alessandro Giacchetto.

• Around the combinatorial unit ball of measured foliations on bordered surfaces, In preparation.

Collaborators

People I have worked with since starting the PhD are:

• Jørgen Ellegaard Andersen
• Gaëtan Borot
• Séverin Charbonnier
• Vincent Delecroix
• Stavros Garoufalidis
• Alessandro Giacchetto
• Danilo Lewanski
• Don Zagier
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