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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).

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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.

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Abstract: The number of irreducible polynomials over a finite field was first counted by Gauss. We will explain a connection between counting the number of irreducible polynomials over F_q with certain properties, and the number of rational points on some related algebraic curves. This idea can be used to count the number of irreducible polynomials with certain coefficients being 0. The appearance of supersingular curves explains the interesting periodic behaviour in the formulae.

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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.

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Past colloquia can be found here.

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