14:00 Dmitrii Mints, Imperial College London
High order homoclinic tangencies and universal dynamics for multidimensional diffeomorphisms
Our research is aimed at studying the dynamics of smooth multidimensional diffeomorphisms from Newhouse domain, that is, open regions in the space of maps where systems with homoclinic tangencies are dense. We prove that in the space of smooth and real-analytic multidimensional maps in any neighborhood of a map such that it has a bi-focus periodic orbit whose invariant manifolds are tangent, there exist open regions (which are subdomain of the Newhouse domain) where maps with high order homoclinic tangencies of corank 2 (invariant manifolds forming the tangency have a plane of common tangent vectors) are dense and maps having universal two-dimensional dynamics are residual.
14:35 Julius Villar, University of Oxford
The incipient infinite cluster for Gaussian level set percolation
Bernoulli percolation is a statistical physics model, first introduced in the 1960s to model the behaviour of a porous medium. This model exhibits a ‘phase transition,’ where its properties change suddenly as a ‘percolation parameter’ crosses a critical value.
The behaviour of the model at the critical value is of great interest to those in the field, and exhibits many peculiarities. For example, connected components in this model can get arbitrarily large, but not infinitely large. Even though the event that there exists an infinitely large component has probability zero, we can formally ‘condition’ on it, by conditioning on a sequence of events which approximate this null event, and taking limits. In the 80’s, Kesten showed that the two most ‘sensible’ approximation schemes give well-defined limits, and moreover that these limits agree. This gives rise to a new model– the ‘incipient infinite cluster model’-- which almost-surely contains an infinite cluster, but otherwise behaves like critical Bernoulli percolation.
In this talk, we show that an analogue of this result holds for planar Gaussian level-set percolation, a continuous model which serves as a generalisation to Bernoulli percolation. We will introduce both models, and the background and motivation behind the problem. If time permits, we will discuss how the original proof from the 80s is adapted to the Gaussian setting, by making use of a ‘white noise’ representation for Gaussian fields.
This talk is based on upcoming joint work with Prof. Dmitry Beliaev
15:20 Philipp Jettkant, Imperial College London Academic
Singular Interactions through Local Times
In this talk, I present a particle system on the positive half-line with reflection at the origin, which arises in the modelling of liquidity in financial systems. The Brownian particles interact through weighted sums of their accumulated local time, driving them towards the reflecting boundary. Depending on the interaction strength, measured by the spectral radius of the weight matrix, this feedback loop can lead to a blow-up of the system in finite time. Despite this singular behaviour, the system is shown to be well-posed. In the second half of the talk, I derive the associated mean-field limit, where blow-ups occur once more in the strong interaction regime. Nonetheless, it is possible to establish propagation of chaos, implying that the blow-up times of the finite system and the mean-field limit synchronise. I conclude by discussing stationary and self-similar solutions of the mean-field limit in the weak feedback regime without blow-ups.
This is based on joint work with Graeme Baker (Columbia University) and Ben Hambly (University of Oxford).
