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.

In this talk, I will report about a joint work with D. Ciubotaru, in which we investigate the Dunkl version of the classical Howe-duality (O(k),spo(2|2)). Similar Fischer-type decompositions were studied before in the works of Ben-Said, Brackx, De Bie, De Schepper, Eelbode, Orsted, Soucek and Somberg for other Howe-dual pairs. Our work builds on the notion of a Dirac operator for Drinfeld algebras introduced by Ciubotaru, which was inspired by the analogous theory for Lie algebras, as well as the work of Cheng and Wang on classical Howe dualities.

I will talk about a way of building graded Lie algebras from certain Heisenberg groups. The input for this construction arises naturally when studying families of algebraic curves, and we'll look at some examples in which Lie theory interacts with number theory in an illuminating way. 

We present several Itô-Wentzell formulae on Wiener spaces for real-valued functionals random field of Itô type depending on measures. We distinguish the full- and marginal-measure flow cases. Derivatives with respect to the measure components are understood in the sense of Lions.
This talk is based on joint work with V. Platonov (U. of Edinburgh), see https://arxiv.org/abs/1910.01892.