Schedule

Welcome

Charlene Kalle  (Universiteit Leiden)

Title: Random intermittent dynamics

Abstract: Intermittent dynamics, where systems irregularly alternate between long periods of different types of dynamical behaviour, has been studied since the work of Pomeau and Manneville in 1980. In random dynamical systems this phenomenon has only been well understood in a few specific cases. A random dynamical system consists of a family of deterministic systems, one of which is chosen to be applied at each time step according to some probabilistic rule. In this talk we will describe the intermittency of some families of random systems with a particular emphasis on how the intermittency of the random system depends on the intermittency of the underlying deterministic systems. This talk is based on joint works with Ale Jan Homburg, Tom Kempton, Valentin Matache, Marks Ruziboev, Masato Tsujii, Evgeny Verbitskiy and Benthen Zeegers. 

Coffee Break

Jonathan Bennett (University of Birmingham)

Title: Adjoint Brascamp-Lieb inequalities

Abstract: The Brascamp-Lieb inequalities are a generalisation of the Hölder, Loomis-Whitney and Young convolution inequalities, and have found many applications in harmonic analysis and elsewhere. We present an "adjoint'' version of these inequalities, which may be viewed as an L^p version of the entropic Brascamp-Lieb inequalities of Carlen and Cordero-Erausquin. As an application we establish some lower bounds on various tomographic transforms such as the classical X-ray transform. This is joint work with Terence Tao.

Lunch

Sabrina Kombrink (University of Birmingham)

Title: Renewal Theory in Dynamics and in Fractal Geometry

Abstract: Renewal theorems have been widely applied to determine the long term behaviour of stochastic systems. Our focus lies on dynamical renewal functions, for which we determine the leading asymptotic term as well as asymptotic terms of lower order. These dynamical renewal theorems can be applied to problems in Geometry. Applications that we are particularly interested in include an analogue of Steiner's formula for fractal sets (concerning an asymptotic expansion of the e-parallel volume as  e→ 0) and the Weyl-Berry conjecture (concerning the eigenvalues of the Dirichlet Laplacian on domains with a fractal boundary). 

Dimitris Gerontogiannis (Universiteit Leiden)

Title: The logarithmic Dirichlet Laplacian on Ahlfors regular spaces

Abstract: This talk introduces the logarithmic Dirichlet Laplacian (log-Laplacian) on metric spaces with a finite Ahlfors regular measure. Some examples of metric spaces admitting such a measure are compact Riemannian manifolds, several fractals, self-similar Smale spaces and the Gromov boundaries of many hyperbolic groups. The log-Laplacian is the logarithmic analogue of the Dirichlet Laplacian on Euclidean spaces. It is intrinsically defined on the metric-measure space, is essentially self-adjoint - even if the manifold has boundary - and has compact resolvent. In fact, the heat semigroup operators are trace class after a critical value. Moreover, it gives rise to a Markov process and can be used to study non-isometric group actions on the underlying space through the prism of noncommutative geometry. This is joint work with Bram Mesland.

Coffee & Collaboration

Tony Samuel (University of Birmingham)

Title: Sturmians, Jarník sets and spectral metrics

Abstract: Sturmian words (subshifts) are combinatorial objects that are quite remarkable just by the fact that they can be formulated in terms of a variety of mathematical frameworks, for instance:

• Billiards - Movement of a ball on a square billiard table.

• Combinations - Aperiodic words that are balanced.

• Geometry - Digitised straight lines or circle rotations.

• Dynamical Systems - Minimal factors.

• Number Theory - Continued fractions and semi-group morphisms.

Given θ = [0; a1, a2, ... ] with unbounded continued fraction entries, we will discuss characterising relations of Sturmian subshifts with slope θ with respect to the regularity properties of spectral metrics as introduced by Kellendonk and Savinien, level sets defined in terms of the Diophantine properties of θ, and complexity notions which are generalisations and extensions of the combinatorial concepts of linearly repetitive, repulsive and power free. This is joint work with M. Gröger, M. Kesseböhmer, A. Mosbach and M. Steffens.

Bram Mesland  (Universiteit Leiden)

Title: Diffusion and the noncommutative geometry of Kleinian groups 

Abstract: In this talk I will explain how harmonic measure theory and the fractional Laplacian on the boundary of the hyperbolic ball naturally fit into the framework of noncommutative geometry. In particular they encode nontrivial K-theoretic information for boundary crossed product C*-algebras by Kleinian groups. 

Coffee Break

Sander Hille  (Universiteit Leiden)

Title: Dynamical systems in spaces of measures: motivation, overview and applications

Abstract: The past decades have shown an increasing interest among mathematicians in dynamical systems in spaces of measures, e.g. formulated through measure-valued differential equations or integral equations related to models for population dynamics. Another natural source of examples is provided by the dynamics defined by Markov operators on probability measures, as encountered in e.g. Iterated Function Systems or random dynamical systems. In this presentation I will discuss motivation for studying such dynamical systems and highlight main results that have been obtained. Such systems link for example stochastic population models in biology with measure-valued solutions to related integral equations. The latter may in turn be approximated by particular differential equations in a space of measures or densities, with which one is more familiar. On the way, a -- far from complete -- overview is sketched with ideas for applications as they can be encountered in the literature and in my mathematical research.

Lunch

Onno van Gaans  (Universiteit Leiden)

Title: Invariant measures for stochastic delay differential equations

Abstract: Stability of autonomous stochastic delay differential equations is often expressed by existence of an invariant measure. Due to the delay, the solution process itself is not a Markov process. Instead, one can consider the process of segments, which takes values in an infinite dimensional state space. In order to apply the Krylov-Bogoliubov method to find an invariant measure, one needs the state space to be a complete separable metric space such that the segment process is (eventually) Feller and such that there exists an initial condition for which the segments are tight. We will consider several choices of state spaces and several ways of showing tightness of solutions. 

Coffee & Collaboration

Conference Dinner

Alexandra Tzella  (University of Birmingham)

Title: Diffusion in arrays of obstacles: beyond homogenisation

Abstract: We examine the diffusion of a chemical or heat released in a homogeneous medium interrupted by an infinite number of impermeable obstacles arranged in a periodic lattice. We extend classical results due to Maxwell, Rayleigh and Keller by applying ideas of large-deviation theory to describe the concentration or temperature distribution at large distances from the point of release. We use matched asymptotics to obtain explicit results in the case of nearly touching obstacles, when the transport is strongly inhibited. The technique developed can be applied to complex systems including porous media and composite materials. This is based on joint work with Y. Farah, D. Loghin and J. Vanneste. 

Collaboration

Special Session - Morning

Richard Sharp (University of Warwick)

Title: A non-symmetric Kesten criterion for random walks on groups

Abstract: A famous result of Kesten from 1959 relates symmetric random walks on countable groups to amenability. Precisely, provided the support of the walk generates the group, the probability of return to the identity in 2n steps decays exponentially fast if and only if the group is not amenable. This led to many analogous “amenability dichotomies”, for example for the spectrum of the Laplacian of manifolds and critical exponents of discrete groups of isometries. I will present a version of the dichotomy for non-symmetric walks. This is joint work with Rhiannon Dougall.

Adarsh Bura (University of Birmingham)

Title: Preservation of shadowing by factor maps 

Abstract: We say a dynamical system has shadowing if every pseudo-orbit can be approximated by a real orbit. Shadowing has been used by Bowen to study Axiom A diffeomorphisms, and by Anosov to study flows on Riemannian manifolds. However, shadowing also has numerous applications, especially in mathematical modelling. Every time a computer iterates a function there is some error, hence pseudo-orbits turn up naturally. In this talk we will look at various notions of shadowing and characterise the properties that the factor maps must have, in order to preserve them. 

Lunch

Olga Maleva (University of Birmingham)

Title: Extreme non-differentiability of typical Lipschitz mappings

Abstract: The classical Rademacher Theorem guarantees that every set of positive measure in a finite-dimensional space contains points of differentiability of every Lipschitz function. A major direction in geometric measure theory research of the last two decades is to explore to what extent this is true for Lebesgue null sets.
In recent joint papers with Dymond we investigate this question from the point of view of differentiability of typical Lipschitz mappings. Here, "typical" is understood in terms of Baire category.
Earlier, we showed that in a set that can be covered by countably many closed purely unrectifiable sets, a typical 1-Lipschitz real-valued function is nowhere differentiable, even directionally. In any other null set, a typical 1-Lipschitz function has many points of differentiability. Our most recent work shows, however, that in any set a typical point, and in "coverable" sets as above every point, is a point of non-differentiability of a typical Lipschitz mapping, scalar or vector-valued, in a strong sense: the derivative ratios at the given point approach every operator of norm at most 1. The result about typical points holds for mappings between Banach spaces with arbitrary norms, while the finite-dimensional result about every point of a "coverable" set is currently proved for a large class of norms but not all, which has led to an interesting problem in combinatorial geometry. 

Michael Dymond (University of Birmingham)

Title: Equivalence classes of separated nets in Euclidean spaces.

Abstract: To what extent may two separated nets in a Euclidean space differ as discrete metric spaces? This question may be approached by considering what control on distortion may be achieved by bijective mappings taking one net to the other. We will discuss various notions of equivalence relation on the class of Euclidean separated nets, describe how these interact and consider questions such as the cardinality of the set of equivalence classes. One of the most important notions is that of bilipschitz equivalence and it was for some time a puzzling open question of Gromov from 1993, whether bilipschitz equivalence distinguishes at all between the separated nets of a Euclidean space. Burago and Kleiner and McMullen (1996) showed that there are in fact bilipschitz non-equivalent nets in every multidimensional space. One result that we present will build on this work and points to a regularity threshold lying in between Lipschitz and Hölder. The talk is based on joint work with Vojtěch Kaluža (IST Austria). 

Coffee & Collaboration