Research

Research interests

Papers and preprints

• Diagonal invariants and genus-zero Hurwitz Frobenius manifolds - preprint available at arXiv:2412.05165 - joint with Ian A. B. Strachan.

Viéta identities famously relate the zeros of a polynomial to the coefficients of the powers of the unknown. Moreover, they realise the polynomial itself as a generating function for the ring of symmetric polynomials. One could ask a similar question for rational functions: provided that they stabilise infinity, performing polynomial division will produce another polynomial, with a rational reminder. The relation between the coefficient of such polynomial and the zeros and poles of the rational function is of interest to us because the space of such functions can be endowed with the structure of a Frobenius manifold. In this setting, the polynomial Vieta case will correspond to the Saito-Dubrovin construction, and elementary symmetric polynomials to Saito flat coordinates. We show that the Frobenius manifold structure on the space of rational function with simple poles is controllable from the one on polynomials, and that the additional information is encoded into a further diagonal action of the symmetric group, whose invariant ring is known to be generated by the polarised power sums.

We, then, perform an analogous extension of the Hurwitz space associated to Dubrovin-Zhang Frobenius manifolds, and show that a similar result holds if one introduces a proper generalisation of the polarised power sums to the trigonometric case. We are able to retrieve a known result by Dafeng Zuo as a special case.

Seminar Talks

• Patterns of limit points of Schottky groups in the plane - Glasgow Postgraduate Seminar, University of Glasgow, February 9th 2024.

Abstract:

Classical Schottky groups are groups generated by Möbius transformations pairing disks in the complex plane. In this talk, we will discuss how moving these disks about influences the properties of the limit set, thus giving rise to some nice-looking patterns in the plane. Incidentally, this is related to a uniformisation result for some surfaces by Schottky groups, which we shall very briefly delve into. No prior knowledge of any of the aforementioned concepts will be assumed, an understanding of how complex multiplication works should (hopefully) be more than enough.

Slides (pictures are either generated by myself or taken from this book).

• Everything's in order in a black hole - Glasgow Postgraduate Seminar, University of Glasgow, March 7th 2025.

Abstract:

Somebody here once said that, when it comes to picking teams for a game of football, I count as an applied Mathematician, as I "do Physics". Having no problem with that, that is what we are doing today. Starting by demystifying the usual punch-line that interpreting gravity as spacetime curvature, rather than as a force, has got anything to do with Relativity, we shall then see what the actual difference between Newton's Mechanics and General Relativity is from a geometrical point of view. In particular, we shall discuss how the notion of causality is implemented in the latter. Finally, by comparing two famous examples of relativistic spacetimes, we are going to describe what conditions may enable the existence of a black hole. If time permits, we shall also talk about rotating black holes and the no-hair Theorem. No prior knowledge of General Relativity or Differential Geometry will be assumed.

• Viéta-like identities for rational functions and Hurwitz Frobenius manifolds - Glasgow ISMP Seminar Series, University of Glasgow, March 18th 2025.

Abstract:

Viéta identities famously relate the zeros of a polynomial to its coefficients.  Our interest in these relations lies in the fact that the space of such functions is naturally a Frobenius manifold. Here, polynomials will play a role in the Saito construction, and Viéta identities in the Saito flat coordinates. We show how this idea can be generalised to spaces of rational functions, and that the corresponding Frobenius manifolds are controllable from polynomial ones. The additional information is encoded into a diagonal action of the symmetric group, whose invariant ring is generated by Weyl's polarised power sums. If time permits, we will see how this applies to Dubrovin-Zhang Frobenius manifolds, and correspondingly to exponential polarised power sums. This is joint with Ian Strachan.

• Permutations of polynomial roots and flat coordinates of Frobenius manifolds - Insalate di Matematica, University of Milan-Bicocca, April 2nd 2025.

Abstract:

A complex polynomial of given degree is completely specified by either its roots or its coefficients. Relations between the two are famously given by Viéta identities. Within the theory of Frobenius manifolds, these serve as a transition function between two distinguished coordinate systems. We discuss how this can be generalised to rational functions. In particular, it turns out that the Frobenius manifold structure on the space of some rational functions can be described from a suitable polynomial one, the additional information being encoded into polynomial invariants under the diagonal action of the symmetric group. This is based on a joint work with Ian Strachan.

Poster and recording available on the Insalate di Matematica website.