University Teacher - The University of Liverpool

Room 424

Department of Mathematical Sciences

Peach Street

Liverpool

L69 7ZL

Singularity theory, monodromy groups, braid groups.

I enjoy giving talks about my research at international conferences, as well as regularly giving popular talks about recreational maths at school and maths clubs. This map shows some of my recent talks.

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[Mathjax] - The best way to typeset mathematics for the web

[IMA] - The Institute of Mathematics and its Applications

[Imaginary] - Visualisation of Algebraic Surfaces

In this work we examine the problem of integral hypercuboids for which the internal core
and the external shell are of equal volume. That is, the volume of the entire hypercuboid
is double that of the internal shell. We will call such a hypercuboid a doubling hypercuboid.
The problem is well known in two and three dimensions. In this article, we have extended the treatment
to include higher (and lower) dimensions.
We have used a computer program to solve the program in four and five dimensions. We show
that the number of solutions in any given dimension is finite, and give some general results
about the smallest and largest possible solutions.

3. Infinite Familes of Monohedral Disk Tilings (not yet published)

__[Abstract]__ - [Preprint - 2011] - [Arxiv - 2015]

Can we construct monohedral tilings of the disk such that a neighbourhood of the origin has trivial intersection with at least one tile? Yes! We present two infinite families, one of which is new.

2. Invariant Symmetries of Unimodal Function Singularities (with Victor Goryunov), Moscow Mathematical Journal, Volume 12 No. 2 (2012) 313-333

__[Abstract]__ - [Preprint] - [Journal Edition]

We classify finite order symmetries \(g\) of the 14 exceptional unimodal function singularities \(f\) in 3 variables, which satisfy a so-called splitting condition. This means that the rank 2 positive subspace in the vanishing homology of \(f\) should not be contained in one eigenspace of \(g_\star\). We also obtain a description of the hyperbolic complex reflection groups appearing as equivariant monodromy groups acting on the hyperbolic eigensubspaces arising.

1. Symmetries of Unimodal Singularities and Complex Hyperbolic Reflection Groups (2011) PhD Thesis

__[Abstract]__ - [Download]

In search of discrete complex hyperbolic reflection groups in a singularity context, we consider cyclic symmetries of the 14 exceptional unimodal function singularities. In the 3-variable case, we classify all the symmetries for which the restriction of the intersection form of an invariant Milnor fibre to a character subspace has positive signature 1, and hence the corresponding equivariant monodromy is a reflection subgroup of \(U(k−1,1)\). For such subspaces, we construct distinguished vanishing bases and their Dynkin diagrams. For \(k=2\), the projectivised hyperbolic monodromy is a triangle group of the Poincaré disk. For \(k=3\), we identify some of the projectivised monodromy groups within a recently published survey by J. R. Parker.