### Programme of the Pure Mathematics Colloquium

The colloquium takes place on Thursdays at 4pm in Theatre C of the Mathematical Institute (unless indicated otherwise).

• 22nd Sept, 2016: Wolfram Bentz (Hull)
Title: The minimal generating sets of the semigroup of transformations stabilising a given partition
Abstract: Let $\mathcal{P}$ be a partition of a finite set $X$. We say that a transformation $f:X\to X$ stabilises the partition $\mathcal{P}$ if for all $P\in \mathcal{P}$ there exists $Q\in \mathcal{P}$ such that $Pf\subseteq Q$. Let $T(X,\mathcal{P})$ denote the semigroup of all full transformations of $X$ that preserve the partition $\mathcal{P}$.

In 2005 Pei Huisheng found an upper bound for the minimum size of the generating sets of $T(X,\mathcal{P})$, when $\mathcal{P}$ is a partition in which all of its parts have the same size. In addition, Pei Huisheng conjectured that his bound was exact. In 2009, Araújo and Schneider used representation theory to solve Pei Huisheng's conjecture.

A more general task is to find the minimum size of the generating sets of $T(X,\mathcal{P})$, when $\mathcal{P}$ is an arbitrary partition. In this talk we presents the solution of this problem and discuss some of the proof techniques, which range from representation theory to combinatorial arguments.

This is joint work with João Araújo (Universade Aberta/CEMAT), James Mitchell (University of St Andrews), and Csaba Schneider (Universidade Federal de Minas Gerais).
• 29th Sept, 2016: Zemer Kosloff (Warwick)
Title: On the Kreiger types of nonsingular Bernoulli shifts
Abstract: A measurable transformation $T$ of a probability space $(\Omega,\mathcal{B},m)$ is quasi invariant if it preserves the $\sigma$-ideal of measure $0$ sets. An old question of Halmos, which was answered in the affirmative by Ornstein and L. Arnold, is whether there exists such transformations which have recurrent dynamics but there exists no $\sigma$-finite, $m$-absolutely continuous $T$ invariant measures. Such systems are called type $III$. The type $III$ transformations can be further classified according to their Krieger types $III_\lambda, 0 \leq \lambda \leq 1$ where being type $III_1$ is equivalent to the Maharam extension being ergodic.

In this talk we will discuss these notions and more in the context of the dynamics of the shift with respect to products measures (not necessarily i.i.d.). If time permits we will discuss an application of these results to symmetric $\alpha$-stable processes, some extensions to the case of the shift of inhomogeneous Markov chain and the construction of a new class of Anosov diffeormorphisms of the torus.
• 6th Oct, 2016: Alla Detinko (St Andrews)
Title: Linear groups and computation
Abstract: In the talk we will survey a novel domain of computational group theory: computing with infinite linear groups. We will provide an introduction to the area, and will discuss available methods and algorithms. Special consideration will be given to the most recent developments in computing with arithmetic groups and its applications. This talk is aimed at a general mathematical audience.
13th Oct, 2016: DOUBLE BILL:
• Christoph Bandt (Greifswald) 3pm in PHYSICS 301 (by the library)
Title: The parametric family of Bernoulli convolutions
Abstract: Bernoulli convolutions are arguably the simplest fractal measures on the unit interval, parametrized by a factor t between 0 and 1. They have been studied for almost 80 years, without much success in the overlapping case. Only for countably many parameters it is exactly known whether the measure admits a density function. We introduce these measures from the viewpoints of probability, fractals, number systems, and dynamical systems. Then we present a new approach which represents the whole parametric family by a function of two parameters. The structure of that function is studied with computer assistance.
• Xiong Jin (Manchester) 4pm in Maths Theatre C
Title: Fractals and probability theory
Abstract: I will talk about some similarities between Fractal Geometry and Probability Theory, in particular the Markov Chains, Random Walks and Iterated Function System. I will then talk about some recent progress on projections and slices of random and deterministic fractal measures on the plane.
• 27th Oct, 2016: Louis Theran (St Andrews)
Title: Generic universal rigidity and the power of SDP for graph realisation
Abstract: A (bar-joint) framework (G,p) is a graph G, along with a placement p of its vertices into R^d. A framework is said to be universally rigid if any other (G,q) in *any dimension* $D\geq d$ that has the same edge lengths as (G,p) is related to (G,p) by a rigid body motion. I'll describe an algebraic characterisation of which graphs G have generic universally rigid frameworks (G,p) and a close connection to a widely used semidefinite programming algorithm for the graph realisation or "distance geometry" problem.

Joint work with Bob Connelly and Shlomo Gortler.
• 3rd Nov, 2016: Sanju Velani (York)
Title: Diophantine approximation in Kleinian groups: singular, extremal and bad limit points
Abstract: The aim is to initiate a manifold'' theory for metric Diophantine approximation on the limit sets of Kleinian groups. We investigate the notions of singular and extremal limit points within the geometrically finite Kleinian group framework. Also, we consider the natural analogue of Davenport's problem regarding badly approximable limit points in a given subset of the limit set. Beyond extremality, we discuss potential Khintchine-type statements for subsets of the limit set. These can be interpreted as the conjectural manifold'' strengthening of Sullivan's logarithmic law for geodesics.
• 10th Nov, 2016: Mirna Djamonja (UEA)
Title: Logical Perspectives of the theory of Graphons
Abstract: Graphons are uncountable limits of sequences of finite graphs. Their invention in 2006 by Lovasz and Szegedy revolutionized both the finite and the infinite graph theory by bringing an unforeseen connection. Graphons, also known as combinatorial limits can be seen as certain ultraproducts, which makes them amenable to study using the methods of logic. We shall give a very general talk about this concept and at the end present some joint results with Tomasic.
• 17th Nov, 2016: Michael Whittaker (Glasgow)
Title: Fractal substitution tilings and applications to noncommutative geometry
Abstract: Starting with a substitution tiling, such as the Penrose tiling, we demonstrate a method for constructing infinitely many new substitution tilings. Each of these new tilings is derived from a graph iterated function system and the tiles typically have fractal boundary. As an application of fractal tilings, we construct an odd spectral triple on a C*-algebra associated with an aperiodic substitution tiling. Even though spectral triples on substitution tilings have been extremely well studied in the last 25 years, our construction produces the first truly noncommutative spectral triple associated with a tiling. My work on fractal substitution tilings is joint with Natalie Frank and Sam Webster, and my work on spectral triples is joint with Michael Mampusti.
• 3pm, 13th Jan, 2017 (NOTE UNUSUAL DAY AND TIME): Pierre-Philippe Dechant (York)
Title: TBA
Abstract: TBA
• 19th Jan, 2017: Jim Belk (Bard)
Title: TBA
Abstract: TBA
Past colloquia can be found here.
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