The study of 'moments' of random matrices (expectations of traces of powers of the matrix) is a rich and interesting subject, with fascinating connections to enumerative geometry, as discovered by Harer and Zagier in the 1980’s. I will give some background on this and then describe some recent work which offers some new perspectives (and new results). This talk is based on joint work with Fabio Deelan Cunden, Francesco Mezzadri and Nick Simm.

We discuss asymptotics of Toeplitz determinants with Fisher--Hartwig singularities, and give an overview of past and more recent results.
Applications include the study of asymptotics of certain statistics of the characteristic polynomial of the Circular Unitary Ensemble (CUE) of random matrices. In particular recent results in the study of Toeplitz determinants allow for a proof of a conjecture by Fyodorov and Keating on moments of averages of the characteristic polynomial of the CUE.
 

In this talk I will give an overview of Seba's billiard as a popular model in the field of Quantum Chaos. Consider a rectangular billiard with a Dirac mass placed in its interior. Whereas this mass has essentially no effect on the classical dynamics, it does have an effect on the quantum dynamics, because quantum wave packets experience diffraction at the point obstacle. Numerical investigations of this model by Petr Seba suggested that the spectrum and the eigenfunctions of the Seba billiard resemble the spectra and eigenfunctions of billiards which are classically chaotic.

I will give an introduction to this model and discuss recent results on quantum ergodicity, superscars and the validity of Berry's random wave conjecture. This talk is based on joint work with Par Kurlberg and Zeev Rudnick.

The triangular inequality is central in Mathematics. What would happen if we reverse it? We only obtain trivial spaces. However, if we mix it with an order structure, we obtain interesting spaces. We'll present linear antimetrics, prove a "masking theorem", and then look at a corollary which tells us about the "twin paradox" in special relativity; time is antimetric!

Background

The RON test is an engine test that is used to measure the research octane number (RON) of a gasoline. It is a parameter that is set in fuels specifications and is an indicator of a fuel to partially explode during burning rather than burn smoothly.

The efficiency of a gasoline engine is limited by the RON value of the fuel that it is using. As the world moves towards lower carbon, predicting the RON of a fuel will become more important.

Typical market gasolines are blended from several hundred hydrocarbon components plus alcohols and ethers. Each component has a RON value and therefore, if the composition is known then the RON can be calculated. Unfortunately, components can have antagonistic or complimentary effects on each other and therefore this needs to be taken into account in the calculation.

Several models have been produced over the years (the RON test has been around for over 60 years) but the accuracy of the models is variable. The existing models are empirically based rather than taking into account the causal links between fuel component properties and RON performance.

Opportunity

BP has developed intellectual property regarding the causal links and we need to know if these can be used to build a functional based model. There is also an opportunity to build a better empirically based model using data on individual fuel components (previous models have grouped similar components to lessen the computing effort)

I will report on recent joint work with Karen Vogtmann on the Euler characteristic of $Out(F_n)$ and the moduli space of graphs. A similar study has been performed in the seminal 1986 work of Harer and Zagier on the Euler characteristic of the mapping class group and the moduli space of curves. I will review a topological field theory proof, due to Kontsevich, of Harer and Zagier´s result and illustrate how an analogous `renormalized` topological field theory argument can be applied to $Out(F_n)$.

We present a programme towards a combinatorial language for higher (stratified) Morse-Cerf theory. Our starting point will be the interpretation of a Morse function as a constructible bundle (of manifolds) over R^1. Generalising this, we describe a surprising combinatorial classification of constructible bundles on flag foliated R^n (the latter structure of a "flag foliation” is needed for us to capture the notions of "singularities of higher Morse-Cerf functions" independently of differentiable structure). We remark that flag foliations can also be seen to provide a notion of directed topology and in this sense higher Morse-Cerf singularities are closely related to coherences in higher category theory. The main result we will present is the algorithmic decidability of existence of mutual refinements of constructible bundles. Using this result, we discuss how "combinatorial stratified higher Morse-Cerf theory" opens up novel paths to the computational treatment of interesting questions in manifold topology.

The tunnel number of a graph embedded in a 3-dimensional manifold is the fewest number of arcs needed so that the union of the graph with the arcs has handlebody exterior. The behavior of tunnel number with respect to connected sum of knots can vary dramatically, depending on the knots involved. However, a classical theorem of Scharlemann and Schultens says that the tunnel number of a composite knot is at least the number of factors. For theta graphs, trivalent vertex sum is the operation which most closely resembles the connected sum of knots. The analogous theorem of Scharlemann and Schultens no longer holds, however. I will provide a sharp lower bound for the tunnel number of composite theta graphs, using recent work on a new knot invariant which is additive under connected sum and trivalent vertex sum. This is joint work with Maggy Tomova.

I will present recent results concerning a class of nonlinear parabolic systems of partial differential equations with small cross-diffusion (see doi.org/10.1051/m2an/2018036 and arXiv:1906.08060). Such systems can be interpreted as a perturbation of a linear problem and they have been proposed to describe the dynamics of a variety of large systems of interacting particles. I will discuss well-posedness, regularity, stability and convergence to the stationary state for (strong) solutions in an appropriate Banach space. I will also present some applications and refinements of the above-mentioned results for specific models.