Research

We study the intersection theory of the Θʳˢ-classes, where r ≥ 2 and 1 ≤ s ≤ r−1, which are cohomological field theories arising as the top-degree parts of Chiodo classes. We show that generalised topological recursion on the (r,s) spectral curves computes the descendant integrals of the Θ-classes. Consequently, we prove that the descendant potential of the Θ-classes is a tau function of the r-KdV hierarchy, generalising the Brézin–Gross–Witten tau function (the special case r = 2, s = 1). We also explicitly compute the W-constraints satisfied by the descendant potential, given by differential representations of the W(gl(r))-algebra at self-dual level. This work extends previously known results on the Witten r-spin class, the r-spin Θ-classes (corresponding to s = r−1), and the Norbury Θ-classes (the case r = 2, s = 1).

We show that the n-point, genus-g correlation functions of topological recursion on any regular spectral curve with simple ramifications grow at most like (2g - 2 + n)! as g tends to infinity, which is the expected growth rate. This provides, in particular, an upper bound for many curve counting problems in large genus and serves as a preliminary step for a resurgence analysis.

We define F-topological recursion (F-TR) as a non-symmetric version of topological recursion, which associates a vector potential to some initial data. We describe the symmetries of the initial data for F-TR and show that, at the level of the vector potential, they include the F-Givental (non-linear) symmetries studied by Arsie, Buryak, Lorenzoni, and Rossi within the framework of F-manifolds. Additionally, we propose a spectral curve formulation of F-topological recursion. This allows us to extend the correspondence between semisimple cohomological field theories (CohFTs) and topological recursion, as established by Dunin-Barkowski, Orantin, Shadrin, and Spitz, to the F-world. In the absence of a full reconstruction theorem à la Teleman for F-CohFTs, this demonstrates that F-TR holds for the ancestor vector potential of a given F-CohFT if and only if it holds for some F-CohFT in its F-Givental orbit. We turn this into a useful statement by showing that the correlation functions of F-topological field theories (F-CohFTs of cohomological degree 0) are governed by F-TR. We apply these results to the extended 2-spin F-CohFT. Furthermore, we exhibit a large set of linear symmetries of F-CohFTs, which do not commute with the F-Givental action.

How can transformers learn and model enumerative geometry? How can they assist in mathematical discovery? In this work, we introduce a new approach to computational enumerative geometry, focusing on ψ-class intersection numbers on the moduli space of curves. We formulate the problem as a continuous optimisation task and develop a transformer-based model leveraging the underlying Virasoro constraints. Our model accurately predicts intersection numbers across a vast range, from 10⁻⁴⁵ to 10⁴⁵. To capture their recursive behaviour, we propose a novel activation function, the Dynamic Range Activator (DRA). Addressing the heteroscedasticity and precision demands of ψ-class intersections, we quantify uncertainty using conformal prediction with a dynamic sliding window. Beyond computations, we investigate the interpretability of the model, showing that transformers implicitly learn Virasoro constraints in a data-driven manner. Finally, we uncover insights into the large-genus asymptotics of ψ-class intersections through abductive hypothesis testing.

We study the length of short cycles on a uniformly random metric map (also known as ribbon graph) of large genus using a Teichmüller theory approach. We establish that, as the genus tends to infinity, the length spectrum converges to a Poisson point process with an explicit intensity. This result extends the work of Janson and Louf to the multi-faced case.

We present a novel approach for computing the large genus asymptotics of intersection numbers. Our strategy is based on a resurgent analysis of the n-point functions of such intersection numbers, which are computed via determinantal formulae, and relies on the presence of a quantum curve. With this approach, we are able to extend the recent results of Aggarwal for Witten–Kontsevich intersection numbers with the computation of all subleading corrections, proving a conjecture of Guo–Yang, and obtain new results on r-spin and Theta-class intersection numbers. 

We study the spin Gromov–Witten (GW) theory of ℙ¹. Using the standard torus action on ℙ¹, we prove that the associated equivariant potential can be expressed by means of operator formalism and satisfies the 2-BKP hierarchy. As a consequence of this result, we prove the spin analogue of the GW/Hurwitz correspondence of Okounkov–Pandharipande for ℙ¹, which was conjectured by J. Lee. Finally, we prove that this correspondence for a general target spin curve follows from a conjectural degeneration formula for spin GW invariants that holds in virtual dimension 0.

We construct and study various properties of a negative spin version of the Witten r-spin class. By taking the top Chern class of a certain vector bundle on the moduli space of stable twisted spin curves, we construct a non-semisimple cohomological field theory that we call the Theta class Θʳ. This CohFT does not have a flat unit and its associated Dubrovin–Frobenius manifold is nowhere semisimple. Despite this, we construct a semisimple deformation of the Theta class, and using the Teleman reconstruction theorem, we obtain tautological relations on the moduli space of stable curves. We further consider the descendant potential of Θʳ and prove that it is the unique solution to a set of W-algebra constraints, which implies a recursive formula for the descendant integrals. Using this result for r=2, we prove Norbury's conjecture which states that the descendant potential of the Theta class coincides with the Brézin–Gross–Witten tau function of the KdV hierarchy. Furthermore, we conjecture that the descendant potential of the Θʳ is the r-BGW tau function of the r-KdV hierarchy and prove the conjecture for r=3. 

We show that the R-matrix and the translation of the shifted Witten classes can be constructed from the solutions of two differential equations that generalise the classical Airy differential equation. Using this, we prove that the descendant intersection theory of the shifted Witten classes satisfies topological recursion on two 1-parameter families of spectral curves and, by taking the limit as the parameter goes to zero, we prove that the descendant intersection theory of the Witten r-spin class is computed by topological recursion on the r-Airy spectral curve. We finally show that this proof suffices to deduce Witten's r-spin conjecture. 

We provide an intersection-theoretic formula for the Euler characteristic of the moduli space of smooth curves. This formula reads purely in terms of Hodge integrals and gives a new short proof of the Harer–Zagier formula. Our result is based on the Gauss–Bonnet formula, and on the observation that a certain parametrisation of the Chiodo class provides the Chern class of the log tangent bundle to the moduli space of smooth curves. 

We study the Lᴾ-integrability of the unit ball's volume of the space of measured foliations on a fixed bordered surface, equipped with an embedded metric ribbon graph. Such volume, studied for the first time by Mirzakhani in the hyperbolic context, appears as the prefactor of the polynomial growth of the number of multicurves. Surprisingly, we found that the maximal p for which the p-norm is finite depends on the topology of the underlying surface, in contrast with the hyperbolic setting where square-integrability has recently been proved. 

We study spin Hurwitz numbers, which count ramified covers of the Riemann sphere with a sign coming from a theta characteristic. These numbers are known to be related to the representation theory of the Sergeev group and the Fock space of type B, from which we derive polynomiality properties of these numbers and we derive a spectral curve which we conjecture computes spin Hurwitz numbers via a new type of topological recursion. We prove that this conjectural topological recursion is equivalent to an ELSV-type formula, expressing spin Hurwitz numbers in terms of the Chiodo class twisted by the 2-spin Witten class. 

We use a combinatorial model for the Teichmüller space to give global Darboux coordinates for the Kontsevich symplectic form on the moduli space of metric ribbon graphs.  We then illustrate how this fits into the framework of geometric recursion, and prove a Mirzakhani–McShane type identity. We further construct an analogue to the Mirzakhani function, which can be explicitly described on the combinatorial moduli space and gives rise to Masur–Veech volumes of the principle stratum of the moduli space of quadratic differentials. 

The authors show how Masur–Veech volumes of the principal strata of quadratic differentials can be computed as intersection numbers arising from the Segre class of the quadratic Hodge bundle, which can be further simplified to certain linear Hodge integrals. In the appendix, we prove that the intersection of this class with ψ-classes can be computed by the Eynard–Orantin topological recursion. 

We prove that Masur–Veech volumes of the principal stratum of the moduli space of quadratic differentials can be calculated by the topological recursion as an application of geometric recursion.