This paper proves quantum modularity of both functions from Q and q-series associated to the closed manifold obtained by −1/2 surgery on the figure-eight knot, 4_1(−1,2). In a sense, this is a companion to work of Garoufalidis-Zagier where similar statements were studied in detail for some simple knots. It is shown that quantum modularity for closed manifolds provides a unification of Chen-Yang's volume conjecture with Witten's asymptotic expansion conjecture. Additionally we show that 4_1(−1,2) is a counter-examples to previous conjectures relating the Witten-Reshetikhin-Turaev invariant and the Zhat(q) series. This could be reformulated in terms of a "strange identity", which gives a volume conjecture for the Zhat invariant. Using factorisation of state integrals, we give conjectural but precise q-hypergeometric formulae for generating series of Stokes constants of this manifold. We find that the generating series of Stokes constants is related to the extension of the 3d index of Dimofte-Gaiotto-Gukov of 4_1(−1,2) proposed by Gang-Yonekura. This extends the equivalent conjecture of Garoufalidis-Gu-Mariño for knots to closed manifolds. This work appeared in a similar form in my thesis.

We prove that a formal power series associated to an ideally triangulated cusped hyperbolic 3-manifold (together with some further choices) is a topological invariant. This formal power series is conjectured to agree to all orders in perturbation theory with two important topological invariants of hyperbolic knots, namely the Kashaev invariant and the Andersen--Kashaev invariant (also known as the state-integral) of Teichmüller TQFT. This is joint work with Stavros Garoufalidis and Matthias Storzer.

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: