It is well-known that the monogenic free inverse semigroup $FI_1$ is not finitely presented as a "plain" semigroup (Schein, 1975). In this talk, I will extend this result and completely characterise when a subsemigroup of $FI_1$ is finitely presented as a semigroup. This is joint work with Nik Ruškuc.

Twisted conjugacy is a generalisation of the well-known notion of conjugacy in groups: given two groups G and H, two homomorphisms φ,ψ: H → G and two elements g, g' in G, we say that g and g' are (φ,ψ)-twisted conjugate if there exists some h in H such that g = ψ(h)⁻¹ g' φ(h). In this seminar, I will discuss three search problems related to twisted conjugacy, and present the current progress towards constructing algorithms that solve these problems for (homomorphisms between) virtually polycyclic groups.

In the 1960's. W. Leavitt studied universal algebras which do not satisfy the \emph{Invariant Basis Number Property (IBN)}. These are algebras that do not have a well-defined rank, that is, algebras for which $R^m\cong R^n$ ($m\neq n$) as $R$-modules which are later called the \emph{Leavitt algebras of module type} $(m,n)$. In 2005, the Leavitt algebra of type $(1,n)$ was found to be the so-called \emph{Leavitt path algebra} of a certain directed graph. 

In the representation theory of finite-dimensional simple Lie algebras $\mathfrak{g}$, two categories of modules stand out due to their contrasting nature. The first is the category of weight modules, consisting of $\mathfrak{g}$-representations where a fixed Cartan subalgebra $\mathfrak{h} \subseteq \mathfrak{g}$ acts semisimply. This category has been extensively studied over the past decades, with a classification of simple modules having finite-dimensional weight spaces obtained by O. Mathieu through the introduction of a special class of modules known as coherent families.

I will give an intro to topological semigroups and the important facts one needs to know about them. I will then discuss a central goal around many of my research projects, that being finding the "best" topology on a given semigroup. I will explain more precisely what that means, give some of the standard techniques in the area with examples, and give some results from my research.

In this talk I will talk about some new results for the escape and
hitting statistics for various open dynamical systems. This includes
1. Poisson limit laws for arbitrary slow mixing hyperbolic billiards,
a connection to RH will be presented too
2. polynomial and exponential escape rates, and
3. where orbits prefer going in the phase space of nonuniformly
expanding dynamical systems. I will outline the idea of the proof for
it, which uses operator renewal theory and generalized Keller-Liverani
perturbation theory.

The game of Rock-Paper-Scissors is an instructive example of cyclic competition between competing populations or strategies in evolutionary biology and game theory, where no single species is an overall winner. Mathematically, this cyclic behaviour can be modelled by ordinary differential equations containing heteroclinic cycles: sequences of equilibria along with trajectories that connect them in a cyclic manner. In simple examples, the equilibria are all similar to each other, in that they all have the same number of non-zero components.

The Submonoid Membership problem in a group G asks, given a finitely generated submonoid M of G and an element g of G, whether g is contained in M. In many other decision problems (such as Subgroup Membership and Rational Subset Membership), decidability is preserved under finite extensions of the group G. In contrast, decidability of Submonoid Membership in G is not known to imply its decidability in finite extensions of G.

The rational numbers have been used to measure quantities since ancient times; however, their implementation in computer languages raises a significant problem: zero has no inverse. To address this issue, J. Bergstra and J. Tucker introduced an algebraic structure called a meadow, which allows for the inversion of zero.