We will be starting the postgraduate forum this year with a series of short talks, introducing the topics on which our PhD students are working. Speaking this year are:

Maria Eugenia Celorrio Ramirez – Arens regularity of Banach algebras

Ai Guan – What is an infinity-algebra?

John Hewetson - Rigidity of frameworks in Euclidean spaces

Zac Hall – Mackey functors and the Burnside ring equivalence

George Liddle - Scaling limits of random growth processes

R. Rado characterised those systems of linear Diophantine equations which are ‘unbreakable' with respect to finite partitions, so that any partition of the positive integers yields a set containing a solution. Similarly, a celebrated theorem of Szemerédi gives rise to a characterisation of linear systems possessing solutions in any ‘dense' set of integers. We discuss variants of these results for certain nonlinear Diophantine equations, and explain how answering these combinatorial questions can lead to improvements in our knowledge of prime numbers.

The concept of negative type was introduced nearly 100 years ago as part of the study of the metric spaces embed isometrically in a Hilbert space: a space embeds isometrically in a Hilbert space if and only if it is of 2-negative type. Enflo introduced the concept of generalized roundness in the 1960s to address a separate question concerning uniform embeddings into Hilbert spaces. It was only in the 1990s that it was shown that the two concepts are in fact equivalent.

It has turned out that the maximal generalized roundness (mgr) of a metric space has many unexpected connections to other properties ofthe space. For example, if a compact Riemannian manifold has maximal generalized roundness at least 1, then the space must be simply connected.

A significant problem is that the mgr is typically very hard to calculate, even for finite metric spaces. Recent interest has been in seeing what can be said for the special class of metric spaces generated by the path metric on a connected graph. A relatively recent result of Sanchez suggests that one can at least numerically calculate the mgr of such spaces, allowing one to perform some experimental work in this area. In practice, this turned out to be much more difficult than expected, but it has allowed us to discover some interesting new `facts', not all of which yet have a proof!

This is joint work with Raveen de Silva.

Many natural processes can be viewed as a form of growth or aggregation, where the growth rate at any given point is a random variable which depends on the geometry of the growing body, usually locally. We will look at a number of mathematical models of random growth which are defined on discrete lattices, and then will discuss some of the limitations of these models. In particular, we will look at the problem of finding a scaling limit for a model, and whether the limit of a discrete model can be viewed as a realistic description of a physical process. The ALE(\(\alpha, \eta\)) family of random growth models use conformal mapping techniques in \(\mathbb{C}\) to avoid some of the problems with models in discrete spaces, and to access powerful tools from complex analysis. We will define this family and discuss what happens to these models for large negative values of the attachment parameter \(\eta\), and justify the name "shy growth models". The scaling limit of the ALE model in this regime will turn out to be the celebrated Schramm-Loewner equation, which we will describe some properties of.

Research talk

Groups have so many varied properties that they can be difficult to deal with in an abstract sense. To deal with this, we embed the group we are interested in into one which adheres to more properties, then we can discuss problems in greater generality than we did before. In this talk we will discuss how to do this in a careful and apt manner, to ensure as little information is lost as possible.

Research talk

Model categories provide a setting for doing homotopy theory in a general setting, and are an important tool in algebraic topology. A model category is defined as a category with three distinguished classes of maps satisfying certain axioms; it is cofibrantly generated when these distinguished classes can be built out of sets of maps. This talk will introduce these concepts in an elementary context, and give some classical examples of cofibrantly generated model categories. If time permits, we will also look at some examples arising from recent work. No prior familiarity with category theory will be assumed.

Each quiver appearing in a seed of a skew-symmetric cluster algebra determines a corresponding group, which we call a cluster group, which is defined via a presentation. Grant and Marsh showed that, for quivers appearing in skew-symmetric cluster algebras of finite type, the associated cluster groups are isomorphic to finite reflection groups and thus are finite Coxeter groups. There are many well-established results for Coxeter presentations and it is natural to ask whether the cluster group presentations possess comparable properties. I will define a cluster group associated to a cluster quiver and explain how the theory of cluster algebras forms the basis of research into cluster groups. As for Coxeter groups, we can consider parabolic subgroups of cluster groups. I will outline a proof which shows that, in the type A case, there exists an isomorphism between the lattice of subsets of the defining generators of the cluster group and the lattice of its parabolic subgroups.

Research talk

The Fell topology is the topology defined by Fell in his 1965 paper "The Dual Spaces of Banach algebras". In this talk, we will define what representations are for Banach algebras, briefly describe how they compare to those of groups and C*-algebras, and what it means for them to be irreducible, in order to define the dual space. There will be a few examples, including one currently being worked on, and we will explore the process of how we go about proving statements about such examples.

Let F be a field of characteristic zero, which admits a biquadratic extension. We give an example of a torus G over F such that its classifying stack is stably rational and {BG}{G} is not equal to 1 in the Grothendieck ring of F-stacks K_0(Stacks_F). This allows us to give an example of a finite group scheme A over F such that BA is stably rational but {BA} is not equal to 1 in K_0(Stacks_F).

We will be introducing graph edge colouring on multigraphs. A recent paper by Chen, Jing and Zang proved the long-standing Goldberg conjecture, and we will be looking at some of the tools and methods used in their proof.

We analyze the asymptotic relative size of the largest independent set of a random d-regular graph on n → ∞ vertices. This problem is very different depending on d because of a surprising phase transition. This is somewhat similar to finding the density of ``water'' above and below its freezing point. These phase transitions are related to algorithmic thresholds, mixing properties, counting, graph reconstruction, graph limits and other questions. We are still far from a complete understanding of all these questions. Our tools are partially coming from statistical physics.

A framework is a combination of a graph and a realisation of its vertices in some space. Given a framework, one natural question to ask is whether this framework is rigid or flexible. We will discuss the complementary notions of continuous and infinitesimal rigidity, and investigate what conclusions we can draw about the rigidity of frameworks in a general sense. Certain families of graphs, namely (2,k)-tight graphs for various k, are particularly relevant to the study of rigid frameworks and so some of the key properties of these graphs will also be examined.

Classical ergodic theory is concerned primarily with the study of a single measure-preserving transformation. If one instead considers collections of transformations, then one is lead to study actions of groups on probability spaces, a topic now known as measured group theory. Two important quantities that arise in this study are the cost and L^2 Betti numbers of a group. Gaboriau (2000) conjectured that these quantities are related to one another by the equation Cost = 1 + (first L^2 Betti number). For infinite Kazhdan groups, a theorem of Bekka and Vallette (1997) states that the first L^2 Betti number is zero, so that the conjecture predicts them to have cost 1. In this talk, I will introduce cost and Kazhdan groups from a probabilistic perspective, and apply this perspective to prove that Kazhdan groups do indeed have cost 1. Joint work with Gabor Pete (Renyi Institue/BME).

Research talk

We show that any two birational projective Calabi-Yau manifolds have the same small quantum cohomology algebras. The key tool used is a version of an algebra called symplectic cohomology, which is built using certain Floer homology groups. Morally, the idea of the proof is to show that both small quantum products are identical deformations of the same symplectic cohomology algebra.