A definition of descendants at one point in graph calculus

We study the genus expansion of Barannikov-Kontsevich solutions of the WDVV equation. In terms of the related graph calculus we give a definition of descendants at one point and prove that this definition satisfies the topological recursion relations in genera 0, 1, and 2, string and dilaton equations, and the pull-back formula.Comment: 18 page

Combinatorics of binomial decompositions of the simplest Hodge integrals

We reduce the calculation of the simplest Hodge integrals to some sums over decorated trees. Since Hodge integrals are already calculated, this gives a proof of a rather interesting combinatorial theorem and a new representation of Bernoulli numbers.Comment: 16 page

Changes of variables in ELSV-type formulas

In [5] I.P. Goulden, D.M. Jackson, and R. Vakil formulated a conjecture relating certain Hurwitz numbers (enumerating ramified coverings of the sphere) to the intersection theory on a conjectural Picard variety. We are going to use their formula to study the intersection theory on this variety (if it is ever to be constructed) by methods close to those of M. Kazarian and S. Lando in [7]. In particular, we prove a Witten-Kontsevich-type theorem relating the intersection theory and integrable hierarchies. We also extend the results of [7] to include the Hodge integrals over the moduli spaces, involving one lambda-class.Comment: 25 pages. Final versio

Intersections in genus 3 and the Boussinesq hierarchy

In this note we prove that the enlarged Witten's conjecture is true in the case of the Boussinesq hierarchy for correlators in genus 3 with descendants only at one point

Belorousski-Pandharipande relation in dGBV algebras

We prove that the genus expansion of solutions of the WDVV equation constructed from dGBV algebras satisfy the differential equation determined by the Belorousski-Pandharipande relation in cohomology of the moduli space of curves $\bar{\mathcal{M}}_{2,3}$.Comment: 22 page

On deformations of quasi-Miura transformations and the Dubrovin-Zhang bracket

In our recent paper we proved the polynomiality of a Poisson bracket for a class of infinite-dimensional Hamiltonian systems of PDE's associated to semi-simple Frobenius structures. In the conformal (homogeneous) case, these systems are exactly the hierarchies of Dubrovin-Zhang, and the bracket is the first Poisson structure of their hierarchy. Our approach was based on a very involved computation of a deformation formula for the bracket with respect to the Givental-Y.-P. Lee Lie algebra action. In this paper, we discuss the structure of that deformation formula. In particular, we reprove it using a deformation formula for weak quasi-Miura transformation that relates our hierarchy of PDE's with its dispersionless limit.Comment: 21 page

The spectral curve and the Schroedinger equation of double Hurwitz numbers and higher spin structures

We derive the spectral curves for $q$-part double Hurwitz numbers, $r$-spin simple Hurwitz numbers, and arbitrary combinations of these cases, from the analysis of the unstable (0,1)-geometry. We quantize this family of spectral curves and obtain the Schroedinger equations for the partition function of the corresponding Hurwitz problems. We thus confirm the conjecture for the existence of quantum curves in these generalized Hurwitz number cases.Comment: 15 pages, journal publication versio