The parabolic algebra \(Ap\) is the weakly closed algebra on the Hilbert Space of square integrable functions on the real line, generated by the unitary semigroup of right translations and the unitary semigroup of multiplication operators by the analytic exponential functions \(e^{i\lambda x}\), \(\lambda \geq 0\). This operator algebra is reflexive, with lattice \(LatAp\) naturally homeomorphic to the unit disc (Katavolos and Power, 1997). In this talk, we explore further properties for \(Ap\) and we determine associated nonselfadjoint algebras derived from isometric representations and from compact perturbations.

The diffusion-limited aggregation process (DLA) was introduced in 1981 as a model for the growth of solid aggregates of multiple particles. In the model, particles on a lattice are started "at large distance" from the existing aggregate and perform a random walk, and are attached when they first become adjacent to a particle already in the aggregate. Physicists were initially interested in the model because it showed long-range correlations despite only involving short-range forces. There are many other open problems to this day, such as to find the scaling limit or even just the growth rate of the cluster. Since its initial definition, DLA has remained of interest to mathematicians because of its conjectured fractal behaviour, and because it has been so difficult to prove any rigorous results.

In this talk, we will first do some of the work required to make the definition precise (including formally starting our particles "at infinity" to avoid making a choice about what "at large distance" means), and then discuss some of the notions of dimension and growth rate which have been used heavily in analysing simulations of DLA and heuristic arguments for some of its properties. We will then discuss one of the few rigorous results known about DLA: an upper bound on its growth rate (and so a lower bound on its "dimension") due to Harry Kesten. If time permits, we will also discuss some of the ideas used in Kesten's proof.

Given a Banach algebra \(A\), we can define the two Arens products on its bidual \(A’’\). Arens introduced these two products in 1951. The space \(A’’\) is a Banach algebra with respect to both products and \(A\) is a closed subalgebra of both these algebras.

In this talk, we will speak about the notions of Arens regularity and strong Arens irregularity and some applications to weighted semigroup algebras.

Random growth occurs in many real world settings, for example, we see it exhibited in the growth of tumours, bacterial growth and lightning patterns. As such we would like to be able to model such processes to determine their behaviour in their scaling limits. Well studied models to describe these different processes include the Eden model and DLA. In a 1998 paper, Hastings and Levitov introduced a one parameter family of conformal maps \(HL(\alpha)\) which can be used to model Laplacian growth processes and allows us to vary between the previous models by varying the parameter \(\alpha\). For the first part of the talk we will consider a regularised version of this model and analyse its scaling limits and fluctuations for different values of \(\alpha\) under capacity rescaling. Then in the second part of the talk we will discuss the anisotropic version of the model and consider the scaling limit in this case.

\(A\)-infinity algebras are a generalisation of associative algebras that were invented by Stasheff in the early sixties to study algebraic structures on loop spaces of topological spaces. Since then, \(A\)-infinity algebras and other "infinity algebras" have found many new applications in areas such as topology, geometry, homological algebra and representation theory. This talk will be a general exposition of these ideas.

A framework \((G, p)\) is the combination of a graph \(G\) and a map \(p\) which realises the vertices of \(G\) within some space. When considering these objects it is natural to ask whether they are "rigid" or "flexible". In order to better understand (generic) rigidity in a particular context, one idea is to try and equate (generic, minimal) rigidity with a graph being in a certain family and this often leads to the appearance of \((k, l)\)-tight graphs. Being able to recursively construct these families of graphs has been crucial to solving numerous problems in rigidity, and this talk will focus on some of the methods and difficulties involved in such a construction.

The present talk is aimed to introduce the basic theory of vertex superalgebras. Therefore, we recall the axioms of a vertex superalgebra to expose two further structures which are naturally considered on it: the conformal and the unitary ones. The application of the vertex superalgebra theory to other areas of mathematics are multiple. As a matter of fact, the result presented in this talk are motived by their connection to chiral conformal field theory, albeit they are intrinsically of general interest. In this framework, such results are part of a joint work with S. Carpi and R. Hillier, aimed to upgrade the correspondence between vertex operator algebras and conformal nets of local von Neumann algebras (stated in a paper of 2018 done by Carpi, Kawahigashi, Longo and Weiner) to the graded-local case.

Every closed subspace of every separable Banach space is the kernel of some bounded operator on the space. We examine some transfinite Banach sequence spaces to see whether or not they have this property.