Given a free group F and a pseudovariety V of finite groups, i.e. a class of finite groups closed under taking subgroups, quotients and finitary direct products, we endow F with its pro-V topology. Given an arbitrary finitely generated subgroup H of F, some classical topological decidability problems are: Is it decidable whether H is dense? Is it decidable whether H is closed?  

We study a class of one-dimensional full branch maps admitting two indifferent fixed points as well as critical points and/or unbounded derivative. Under some mild
assumptions we prove the existence of a unique invariant mixing absolutely continuous
probability measure, study its rate of decay of correlation and prove a number of limit
theorems.

What is the expected number of iterates of a point needed for a plot of these iterates to approximate the attractor of the dynamical system up to a given scale delta (i.e., the orbit will have visited a delta-neighbourhood of every point in the attractor)?  This question has analogues in random walks on graphs and Markov chains and can be seen as a recurrence problem.  I'll present joint work with Natalia Jurga (St Andrews) where we estimate the expectation for this problem as a function of delta for some classes of interval maps using ideas from Hitting Time Statistics and transfer operato

Dynamical systems generated by scalar reaction-diffusion equations enjoy special properties that lead to a very simple structure for the semiflow. Among these properties, the monotone behavior of the number of zeros of the solutions plays an essential role. This discrete Lyapunov functional contains important information on the spectral behavior of the linearization and leads to a Morse-Smale description of the dynamical system.

Given a group $G$ acting on a set $X$, an element $g$ of $G$ is called a derangement if it acts without fixed points on $X$. The Boston-Shalev conjecture, proved by Fulman and Guralnick, asserts that in a finite simple group $G$ acting transitively on $X$, the proportion of derangements is at least some absolute constant $c>0$. After giving an introduction to the subject, I will present a version of the conjecture for the proportion of   *conjugacy classes* containing derangements in finite groups of Lie type.

Let $H$ be a monoid (written multiplicatively). We call $H$ Archimedean if, for all $a, b \in H$ such that $b$ is a non-unit, there is an integer $k \ge 1$ with $b^k \in HaH$; strongly Archimedean if, for each $a \in H$, there is an integer $k \ge 1$ such that $HaH$ contains any product of any $k$ non-units of $H$; and duo if $aH = Ha$ for all $a \in H$. For instance, commutative monoids are duo and numerical monoids are strongly Archimedean.

Just as groups are algebraic models of symmetry, inverse semigroups capture the notion of partial symmetry. One of the major results in algebraic semigroup theory is the ESN (Ehresmann-Schein-Nambooripad) Theorem which describes the structure of inverse semigroups using groupoids. In this talk, we discuss how we can generalise this to the class of regular $\star$-semigroups. Just like inverse semigroups, regular $\star$-semigroups have an involution to capture the symmetry, but their idempotents no longer form a semilattice.

It is well-known that the subword complexity of an automatic sequence grows at most linearly, meaning that the number of length-$\ell$ subwords which appear in a given automatic sequence $a = (a(n))_n$ is at most $C \ell$ for a constant $C$ dependent only on $a$. In contrast, arithmetic subword complexity measures the number of subwords which appear along arithmetic progressions, and can grow exponentially even for very simple automatic sequences.

In this talk, we will study a gauge-theoretic construction of (branched) complex projective structures on a closed Riemann surface X of genus g>1. This construction underlies the celebrated conformal limit of Gaiotto and provides a preliminary understanding of its geometry.  More concretely, the non-abelian Hodge correspondence maps a polystabl SL(2,R)-Higgs bundle on X to a connection which, in some cases, is the holonomy of a branched hyperbolic structure.

The length of a finite system of generators for a finite-dimensional algebra over a field is the least positive integer $k$ such that the products of length not exceeding $k$ span this algebra as a vector space.The maximum length for the systems of generators of an algebra is called the length of this algebra. Length function is an important invariant widely used to study finite dimensional algebras since 1959. The length evaluation is a difficult problem.  For example, the length of the full matrix algebra is still  unknown.