Research

Preprint available at arXiv:2412.05165.

Abstract:

The Frobenius manifold structure on the space of rational functions with multiple simple poles is constructed. In particular, the dependence of the Saito-flat coordinates on the flat coordinates of the intersection form is studied. While some of the individual flat coordinates are complicated rational functions, they appear in the prepotential in certain combinations known as diagonal invariants, which turn out to be polynomial. Two classes are studied in more detail. These are generalisations of the Coxeter and extended-affine-Weyl orbits space for the group W=W(A_l). An invariant theory is also developed.

Abstract:

Frobenius manifolds were introduced by B. Dubrovin in the nineties as a geometrical way to encode solutions to the WDVV equations arising in 2d topological field theories. The parameter spaces of miniversal deformations of ADE singularities are well-known to carry such a structure. Much more recently, the existence of an extension to a flat F-manifold of one dimension higher was shown in these cases. We argue that extensions of rank two should be expected, even though we are yet to find them. This is based on ongoing work with my supervisor Ian Strachan.

Abstract:

It is a common phenomenon that Landau-Ginzburg models for Frobenius manifolds coming from extensions of Weyl groups of a fixed Dynkin type are related to one another by suitably changing the ramification profile over infinity of the superpotential, while fixing the primary form. In this talk, we discuss this problem more systematically in type-A. We show that, starting from the polynomial Saito Frobenius manifolds, one can describe the structure on any Hurwitz space such that the additional poles of the superpotential are all simple. The corresponding solution to the WDVV equations exhibits a peculiar invariance under a diagonal action of the symmetric group. The result relies on a novel approach to the computation of flat coordinates according to some suitable embeddings of Hurwitz spaces. This is joint work with Ian Strachan.

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.

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.

Poster

Abstract:

Somebody here once said that, when it comes to picking teams for 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.

Abstract:

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).