I will explain what the Willmore Morse Index of unbranched Willmore spheres in Euclidean three-space is and how to compute it. It turns out that several geometric properties at the ends of complete minimal surfaces with embedded planar ends are related to the mentioned Morse index.
One consequence of that computation is that all unbranched Willmore spheres are unstable (except for the round sphere). This talk is based on work with Jonas Hirsch.

 

We build on the literature on financial contagion using models of cross-holdings of equity participations and debt in different seniority classes, and extend them to include bail-ins and contingent convertible debt instruments, two mechanisms of debt-to-equity conversion. We combine these with recently proposed methods of network valuation under stochastic external assets, allowing for the pricing of debt instruments in each seniority layer and the calculation of default probabilities. We show that there exist well-defined valuations for all financial assets cross-held within the system. The full model constitutes an extension of classic asset pricing models that accounts for cross-holdings of debt securities. Our contribution is to add convertible debt to this framework.

This talk will explain how Lie polynomials underpin the structure of the so-called double copy relationship between gauge and gravity theories (and a network of other theories besides).  ABHY have recently shown that Lie polynomials arise naturally also in the geometry of the space K_n of momentum invariants, Mandelstams, and can be expressed in the space of n-3-forms dual to certain associahedral (n-3)-planes. They also arise in the moduli space M_{0,n} of n points on a Riemann sphere up to Mobius transformations in the n-3-dimensional homology.  The talk goes on to give a natural correspondendence between K_n and the cotangent bundle of M_{0.n} through which the relationships of some of these structures can be expressed.  This in particular gives a natural framework for expressing the CHY and ambitwistor-string formulae for scattering amplitudes of gauge and gravity theories and goes some way to expressing their double copy relations.   This is part of joint work in progress with Hadleigh Frost.

Finitely presented groups are a natural algebraic generalisation of the collection of finite groups. Unlike the finite case there is almost no hope of any kind of classification.

The goal of random groups is therefore to understand the properties of the "typical" finitely presented group. I will present a couple of models for random groups and survey some of the main theorems and open questions in the area, demonstrating surprising correlations between these probabilistic models, geometry and analysis.

I'll give a general introduction to tensor-triangular geometry, the algebraic study of tensor-triangulated categories as they appear in topology, geometry and representation theory. Then I'll discuss an elementary idea, that of a "field" in this theory, and explain what we currently know about them.