Mondays at 2pm
Frank Adams 1, Alan Turing Building
Contacts Donald Robertson Yotam Smilansky
A mild mixing rank one system with non-trivial joinings
Donald Robertson
University of Manchester
In recent work with Jon Chaika an ergodic measure-preserving system was constructed having the properties in the title, answering a question of Thouvenot. In this talk I will describe the approach taken to construct such a system, which is different from the usual cutting and stacking procedure, and outline how this relates to other questions about rank one systems.
TBD
Chay Paterson
University of Manchester
TBD
Andrew Mitchell
The Open University
TBD
Xiong Jin
University of Manchester
TBD
Daniel Meyer
University of Liverpool
TBD
Leticia Pardo-Simón
Universitat de Barcelona
TBD
Anna Jove Campabadal
Universitat de of Barcelona
TBD
Phil Rippon & Gwyneth Stallard
The Open University
An inverse theorem for the Gowers U^3 norm relative to quadratic level sets
Sean Prendiville
University of Lancaster
The Gowers uniformity norms have become well-used tools in additive combinatorics, ergodic theory and analytic number theory. We discuss the utility of these objects and their associated inverse theorems. We report on a recent effective version of the inverse theorem for the Gowers U^3-norm for functions supported on high-rank quadratic level sets in finite vector spaces. This enables one to run so-called "density increment arguments" with respect to quadratic level sets, which are analogues of Bohr sets in the context of quadratic Fourier analysis on finite vector spaces. For instance, one can use such an approach to derive an exponential bound on the Ramsey number of three-term progressions which are the same colour as their common difference ("Brauer quadruples"), a result it seems difficult to obtain by other means.
Fourier asymptotics and effective equidistribution
Shreyasi Datta
University of York
We talk about effective equidistribution of the expanding horocycles on the unit cotangent bundle of the modular surface with respect to various classes of Borel probability measures on the reals, depending on their Fourier asymptotics. This is a joint work with Subhajit Jana based on https://arxiv.org/abs/2407.11961.
Zeros of polynomials with restricted coefficients: a problem of Littlewood
Benjamin Bedert
University of Oxford
The study of polynomials whose coefficients lie in a given set $S$ (the most notable examples being $S=\{0,1\}$ or $\{-1,1\}$) has a long history leading to many interesting results and open problems. We begin with a brief general overview of this topic and then focus on the following old problem of Littlewood. Let $A$ be a set of positive integers, let $f_A(x)=\sum_{n\in A}\cos(nx)$ and define $Z(f_A)$ to be the number of zeros of $f_A$ in $[0,2\pi]$. The problem is to estimate the quantity $Z(N)$ which is defined to be the minimum of $Z(f_A)$ over all sets $A$ of size $N$. We discuss recent progress showing that $Z(N)\geqslant (\log \log N)^{1-o(1)}$ which provides an exponential improvement over the previous lower bound. A closely related question due to Borwein, Erdelyi and Littmann asks about the minimum number of zeros of a cosine polynomial with $\pm 1$-coefficients. Until recently it was unknown whether this even tends to infinity with the degree $N$. We also discuss work confirming this conjecture.
Self-similar groups and their limit spaces
Mike Whittaker
University of Glasgow
A self-similar group (G,X) consists of a group G acting faithfully on a homogeneous rooted tree such that the action satisfies a self-similarity condition. In this talk I will introduce a beautiful theorem of Nekrashevych: there is a self-similar group associated to every post critically finite rational function whose limit space recovers the Julia set of the function. I'll then show how limit spaces of self-similar actions appear in other contexts and reveal dynamical systems in a group theoretic context.
Coincidence and disparity of fractal dimensions for dynamically defined sets
Amlan Banaji
Loughborough University
There are many different ways that one can define fractal dimension. The three of interest in this talk will be Hausdorff dimension, lower box dimension, and upper box dimension. It is well known that these notions of dimension can all be different for general sets, but they coincide for many classes of fractal sets such as self-similar sets. This raises the question of what conditions one needs to assume about an iterated function system (IFS) to be sure that all dimensions will coincide for the resulting fractal? We will survey some results and open problems related to this broad question in three specific settings, namely affine IFSs, bi-Lipschitz IFSs, and infinite conformal IFSs. This talk is based on a joint project with Simon Baker, De-Jun Feng, Chun-Kit Lai and Ying Xiong, and another paper with Alex Rutar.
Comparing separated nets in Euclidean spaces
Michael Dymond
University of Birmingham
Separated nets of Euclidean spaces, sometimes called Delone sets in the literature, are studied in connection with Dynamical Systems, Ergodic Theory and Quasicrystals in Mathematical Physics. We will discuss the question of the extent to which any two separated nets of the same Euclidean space may differ as discrete metric spaces. Our approach to this will be to assess the degree of control, or regularity, with which one separated net may be mapped bijectively to the other. This leads further to the question of extendibility of mappings defined on a separated net, preserving regularity. We present some recent progress on the extension question for bilipschitz mappings of planar separated nets. This is joint work with Vojtěch Kaluža (IST Austria).
Quasimorphisms on the group of area-preserving homeomorphisms on S^2
Yongsheng Jia
University of Manchester
In topological dynamics, the group of orientation-preserving homeomorphisms on the circle has been widely studied. Started by Poincare, people tried to look at the rotation number, which is a map from the group of orientation-preserving homeomorphisms on the circle to the real numbers. It is actually an example of a quasimorphism, which is almost a homomorphism (up to a uniformly bounded error). In this talk, I will focus on how to construct quasimorphisms on the group of area-preserving homeomorphisms on S^2, through a geometric group theoretical point of view, as well some applications to study the large-scale geometry of this group. This is the joint work with my supervisor, Richard Webb.
Projections and sum sets of self-affine fractals
Ian Morris
Queen Mary University
I will describe some recent joint work with Çağrı Sert in which we establish a Falconer-like formula for the dimensions of arbitrary projected images of self-affine fractals. As applications we give examples of self-affine fractals which have large sets of exceptional projections in the sense of Marstrand's theorem; self-affine sets which have small sum sets; and projections and convolutions of natural measures on self-affine sets which fail to be exact-dimensional.
Low Discrepancy Digital Hybrid Sequences and the t-adic Littlewood Conjecture
Steven Robertson
University of Manchester
The discrepancy of a sequence measures how quickly it approaches a uniform distribution. Given a natural number d, any collection of one-dimensional so-called low discrepancy sequences {S_i : 1 ≤ i ≤ d} can be concatenated to create a d-dimensional hybrid sequence (S_1, . . . , S_d). Since their introduction by Spanier in 1995, many connections between the discrepancy of a hybrid sequence and the discrepancy of its component sequences have been discovered. However, a proof that a hybrid sequence is capable of being low discrepancy has remained elusive. In this talk, an explicit connection between Diophantine approximation over function fields and two dimensional low discrepancy hybrid sequences is provided. Specifically, it is shown that any counterexample to the so-called t-adic Littlewood Conjecture (t-LC) can be used to create a low discrepancy digital Kronecker-Van der Corput sequence. Such counterexamples to t-LC are known explicitly over a number of finite fields by, on the one hand, Adiceam, Nesharim and Lunnon, and on the other, by Garrett and the Robertson. All necessary concepts will be defined in the talk.Equivalence relations on aperiodic point sets
Yotam Smilansky
University of Manchester
A Delone set in Euclidean space is an infinite well-spaced (uniformly discrete and relatively dense) set of points. Examples range from perfectly ordered lattices through vertices of aperiodic Penrose tilings to random disordered configurations. A natural question concerns the intermediate behaviour: given a non-lattice Delone set, in what ways is it still "equivalent" to a lattice? I will introduce bounded displacement (BD) and biLipschitz (BL) equivalence relations on Delone sets, and describe fundamental results including the existence of BL-non-equivalent Delone sets (McMullen and Burago-Kleiner, independently, answering a question by Gromov) and criteria for BD and BL equivalence to a lattice (Laczkovich and Burago-Kleiner, respectively), both related to discrepancy estimates. I will then survey more recent advances, with special attention to aperiodic point sets arising through cut-and-project schemes and substitution tilings. No prior knowledge is assumed.A counterexample to Eremenko's conjecture
Lasse Rempe
University of Manchester
Let f be an entire function (i.e., a holomorphic self-map of the complex plane), and suppose that f is transcendental, i.e., not a polynomial. The *escaping set* of f consists of those points that tend to infinity under repeated application of f. (For example, all real numbers belong to the escaping set of the exponential map, since they tend to infinity under repeated exponentiation.) In 1989, Eremenko conjectured that every connected component of the escaping set is unbounded.
Eremenko's conjecture has been a central problem in transcendental dynamics in the past decade. A number of stronger versions of the conjecture have been disproved, while weaker ones has been established, and the conjecture has also been shown to hold for a number of classes of functions. I will describe recent work with David Martí-Pete and James Waterman in which we construct a counterexample to the conjecture. The talk should be accessible to a general mathematical audience, including PhD students.