Research

My research lies in the intersection of mathematical physics and geometry. Below you can find my publications and preprints and a short description of them. 

Hamiltonian formalism for non-diagonalisable systems of hydrodynamic type 

This work has been accepted to Communications of Mathematical Physics. In the meantime you can find the preprint here (joint work with Paolo Lorenzoni and Sara Perletti).

This is the third of a trilogy of papers, joint with Prof. Paolo Lorenzoni and Dr. Sara Perletti, where we extend Tsarev's theory to non-diagonalisable (regular/block-diagonalisable) integrable systems of hydrodynamic type.  

In this paper, we treat the semi-Hamiltonian formalism for regular/block-diagonalisable integrable systems of hydrodynamic type. Using geometric methods, and under an additional restriction which we call the Darboux-Tsarev condition, we study the associated system of conservation laws (Hamiltonian densities) and the Dubrovin-Novikov-Tsarev system for pseudo-Riemannian metrics. We classify the general solution for these systems in terms of a certain number of functional parameters, generalising the semisimple/diagonalisable case. Notably, in the non-semisimple/diagonalisable setting, integrability/compatibility of the system for the symmetries does not automatically lead to the compatibility for the system of the metric. Consequently, the proof of compatibility becomes far more challenging. 

The generalised hodograph method for non-diagonalisable integrable systems of hydrodynamic type 

Preprint can be found here (joint with Paolo Lorenzoni and Sara Perletti).

This is the second of a trilogy of papers, joint with Prof. Paolo Lorenzoni and Dr. Sara Perletti, where we extend Tsarev's theory to non-diagonalisable (regular/block-diagonalisable) integrable systems of hydrodynamic type.  

In this paper we extend the generalised hodograph method to systems introduced above. 

We also study the system of symmetries for integrable systems of hydrodynamic type. We find that under an additional restriction, which we call the Darboux-Tsarev condition, the system for the symmetries may be arranged into a suitable configuration of subsystems for which Darboux theory may be applied. We characterise the general solution in terms of a certain number of functional parameters, naturally generalising the semisimple/diagonalisable case.

Integrable hierarchies and F-manifolds with compatible connection 

This work was published by Communications of Mathematical Physics and can be found here (joint with Paolo Lorenzoni and Sara Perletti).

This is the first of a trilogy of papers, joint with Prof. Paolo Lorenzoni and Dr. Sara Perletti, where we extend Tsarev's theory to non-diagonalisable (regular/block-diagonalisable) integrable systems of hydrodynamic type.  

Building on the interplay between geometry and integrability, we show that (regular) F-manifolds with compatible connection are the geometric counterpart of (block-diagonalisable) integrable systems of hydrodynamic type. We show that, assuming a minor technical assumption, there exists a unique F-manifold with compatible connection and flat unit for a given (block-diagonalisable) system of hydrodynamic type (not necessarily diagonalisable). We give an explicit description of the connection in terms of  Christoffel symbols, and apply the construction to integrable hierarchies  arising from Frolicher-Nijenhius bicomplexes. 

Mirror symmetry for extended affine Weyl groups 

This work was published by Journal de l'École polytechnique — Mathématiques and can be found here (joint Andrea Brini).

In this work we find Landau-Ginzburg models for Dubrovin-Zhang Frobenius manifolds, which are Frobenius manifolds defined as orbit spaces of a certain extension of affine Weyl groups. We do so, in an uniform way, by constructing spectral curves as the characteristic equation of a Lax operator associated to the relativistic Toda chain. We give explicit relations for all Dynkin types and relate them to previously known expressions (for the non-exceptional cases). 

As applications, we give closed-form expressions for the flat coordinates of the Saito metric and the Frobenius prepotentials in all Dynkin types, compute the topological degree of the Lyashko–Looijenga mapping for certain higher genus Hurwitz space strata, and construct hydrodynamic bi-Hamiltonian hierarchies (in both Lax–Sato and Hamiltonian form) that are root-theoretic generalisations of the long-wave limit of the extended Toda hierarchy. 

A Geometric Framework for Pitch Estimation on Acoustic Musical Signals

 This work was published in the Journal of Mathematics and Music and can be found here (joint with Tom Goodman and Peter Tino)

This paper presents a geometric approach to pitch estimation (PE)-an important problem in Music Information Retrieval (MIR), and a precursor to a variety of other problems in the field. Though there exist a number of highly-accurate methods, both mono-pitch estimation and multi-pitch estimation (particularly with unspecified polyphonic timbre) prove computationally and conceptually challenging. A number of current techniques, whilst incredibly effective, are not targeted towards eliciting the underlying mathematical structures that underpin the complex musical patterns exhibited by acoustic musical signals. Tackling the approach from both a theoretical and experimental perspective, we present a novel framework, a basis for further work in the area, and results that (whilst not state of the art) demonstrate relative efficacy. The framework presented in this paper opens up a completely new way to tackle PE problems, and may have uses both in traditional analytical approaches, as well as in the emerging machine learning (ML) methods that currently dominate the literature.