Tau function and moduli of differentials

The tau function on the moduli space of generic holomorphic 1-differentials on complex algebraic curves is interpreted as a section of a line bundle on the projectivized Hodge bundle over the moduli space of stable curves. The asymptotics of the tau function near the boundary of the moduli space of 1-differentials is computed, and an explicit expression for the pullback of the Hodge class on the projectivized Hodge bundle in terms of the tautological class and the classes of boundary divisors is derived. This expression is used to clarify the geometric meaning of the Kontsevich-Zorich formula for the sum of the Lyapunov exponents associated with the Teichm\"uller flow on the Hodge bundle.Comment: misprints corrected; journal reference adde

A hybrid molecular dynamics/fluctuating hydrodynamics method for modelling liquids at multiple scales in space and time

A new 3D implementation of a hybrid model based on the analogy with two-phase hydrodynamics has been developed for the simulation of liquids at microscale. The idea of the method is to smoothly combine the atomistic description in the molecular dynamics zone with the Landau-Lifshitz fluctuating hydrodynamics representation in the rest of the system in the framework of macroscopic conservation laws through the use of a single "zoom-in" user-defined function s that has the meaning of a partial concentration in the two-phase analogy model. In comparison with our previous works, the implementation has been extended to full 3D simulations for a range of atomistic models in GROMACS from argon to water in equilibrium conditions with a constant or a spatially variable function s. Preliminary results of simulating the diffusion of a small peptide in water are also reported

Riemann-Hilbert problem for Hurwitz Frobenius manifolds: regular singularities

In this paper we study the Fuchsian Riemann-Hilbert (inverse monodromy) problem corresponding to Frobenius structures on Hurwitz spaces. We find a solution to this Riemann-Hilbert problem in terms of integrals of certain meromorphic differentials over a basis of an appropriate relative homology space, study the corresponding monodromy group and compute the monodromy matrices explicitly for various special cases.Comment: final versio

Stieltjes–Bethe equations in higher genus and branched coverings with even ramifications

We describe projective structures on a Riemann surface corresponding to monodromy groups which have trivial SL(2) monodromies around singularities and trivial PSL(2) monodromies along homologically non-trivial loops on a Riemann surface. We propose a natural higher genus analog of Stieltjes–Bethe equations. Links with branched projective structures and with Hurwitz spaces with ramifications of even order are established. We find a higher genus analog of the genus zero Yang–Yang function (the function generating accessory parameters) and describe its similarity and difference with Bergman tau-function on the Hurwitz spaces

Periods of meromorphic quadratic differentials and Goldman bracket

We study symplectic properties of monodromy map of second order linear equation with meromorphic potential having only simple zeros on a Riemann surface. We show that the canonical symplectic structure on the cotangent bundle $T^* Mcal_{g,n}$ implies under an appropriately defined monodromy map the Goldman Poisson structure on the corresponding character variety, thereby extending the recent results of the paper of M.Bertola, C.Norton and the author from the case of holomorphic to meromorphic potentials.Non UBCUnreviewedAuthor affiliation: Concordia UniversityFacult