There are a number of features in statistical mechanics which we believe are shared by a large number of models: universality, scale invariance, etc. These properties are often assumed for a model before we have shown rigorously that they hold, and then used to derive useful information. In this talk, I will use the example of Galton-Watson trees and their scaling limit, the continuum random tree, to explain what we mean by universality and scale invariance, and how studying these properties of models can give us information about combinatorial objects such as trees.

The aim of this talk is to give a very general introduction to vertex superalgebra theory. Through a brief historical overview, I will present the connections between this theory and other areas of mathematics and physics: finite groups, modular functions, infinite dimensional Lie algebras, string theory and quantum field theory. Then, I will introduce the axioms of the theory and their consequences, to give a different perspective on the topic. If we have enough time, I will finish with some classical examples and/or a mention of further structures which belong to a vertex superalgebra.

We will introduce the notion of "asymptotic expansion" recently defined by Li, Nowak, Spakula and Zhang and give operator algebraic motivation for its study. We will explore this definition in the familiar setting of Cayley graphs, and show that this leads to a C*-algebraic characterisation of usual expanders for graphs that are sufficiently symmetric. This is joint work with Li, Vigolo and Zhang.

Research talk

The aim of this talk is to introduce the concepts of saturation and weak saturation. Let \(F\) and \(H\) be graphs. We say a graph \(G\) is \((F,H)\)-saturated if \(G\) is a spanning subgraph of \(F\) that: (i) does not contain a copy of \(H\); (ii) for any edge \(e \in E(F) \setminus E(G)\), \(G+e\) contains a copy of \(H\) as a subgraph. A saturation problem aims to determine the value \(\mathrm{sat}(F,H)\), defined as the minimum number of edges for a \((F,H)\)-saturated graph.

Further we shall introduce the definition of weakly saturated graphs. We view examples with complete graphs, complete bipartite graphs, and hypercubes, using this to see constructions that upper bounds for saturation problems can be easy to find. We hope to have a look at ways to find lower bounds to finish the talk.

I’ll report on my recent joint work with Gabor Tardos on convergence and limits of finite trees. The preprint is available on arxiv

We consider wireless communication networks where service rates of nodes depend on their signal to interference plus noise ratios. We demonstrate that these rates are utility-maximising in a certain sense. Using this observation, we prove stability of single-hop networks under natural assumptions. We also obtain bounds on some moments of the stationary workload. In order to obtain these results, we consider a framework of utility-optimising rate allocations which is much more general than the well-known \(\alpha\)-fairness, and prove general results on stability and bounds for some stationary workload distributions in networks with these rate allocations. We also discuss how our results may be applied to the stability analysis of infinite networks. This is joint work with Sasha Stolyar (UIUC).

Research talk

In this talk we will introduce the concept of a mixed identity for a Banach algebra. We will introduce some results regarding the connection of mixed identities and the Arens regularity of a Banach algebra. We will finish the talk with some examples.

Stable module categories of selfinjective algebras are an important class of triangulated categories. The central open problem for stable module categories is the Auslander–Reiten conjecture, which states that stable equivalences preserve the number of isomorphism classes of non-projective simple modules. The study of this conjecture led to the notion of simple- minded systems. This notion generalises the concept of simple modules, which are the natural generators of stable module categories, given the absence of projective modules.

In this talk, we discuss the notion and properties of simple-minded systems in the setting of negative Calabi–Yau triangulated categories. A motivating class of examples of such categories are the stable module categories of symmetric algebras, which are −1 Calabi–Yau. We will also describe a reduction technique for constructing these objects inductively.

This will be a report on joint work with David Pauksztello.

Research talk

Braids groups are quite interdisciplinary: one can find topological, geometrical and algebraic definitions. In this talk we will see that braid groups can be defined as mapping class groups, configuration spaces and also as purely algebraically described groups with a nice structure. We will also learn how braids can connect topological results to algebraic results using geometric group theory. Finally, we will talk about the role of braids in applied domains, such as cryptography or fluid mixing.

A P-graph is a simple graph that embeds in the real projective plane. I will outline the proof of the main result: A (3,6)-tight P-graph is constructible from one of 8 uncontractible "base graphs" by a sequence of vertex splitting moves. It follows from this that a P-graph is minimally generically rigid in 3D if and only if it is (3,6)-tight. This is an exact counterpart of the classical result for the sphere. I shall also indicate general problems for M-graphs for general classical surfaces M. This is joint work with Lefteris Kastis.

In 2013, Choi, Farah and Ozawa constructed the first example of an amenable operator algebra which is not isomorphic to a C-algebra. We will go through this construction and the ideas behind the proof that it isn't isomorphic to a C-algebra. If we have time, we will also discuss briefly work we're currently doing to show that amenable subalgebras of convergent sequences of 2 x 2 matrices are isomorphic to C*-algebras, though the work is very much ongoing.

Mirror symmetry evokes a correspondence between deformation equivalence classes of toric varieties and mutation equivalence classes of the corresponding Fano varieties. The talk gives an insight into a program aimed at classifying Fano varieties using this correspondence.

On Thursday 20th February the LMS Scheme 3 network Algebraic Quantum Field Theory in the UK will hold a one-day meeting in Lancaster. The talks are open to everybody. For more information click here Or ask Robin Hillier.

Research talk

In 2015, B. De Pagter and A. W. Wickstead introduced free Banach lattices as those satisfying a certain universal property of extension of maps defined on a given set. In this extension, bounded maps from a set onto a Banach lattice are upgraded to lattice homomorphisms defined on the free Banach lattice generated by the set. The introduction of free Banach lattices automatically posed new questions about their structure which still remain open. Our aim is to give a mild introduction to free Banach lattices. We will introduce the explicit construction of the free Banach lattice generated by a Banach space, originally delivered by A. Avilés, J. Rodríguez and P. Tradacete. Using it as a motivation, we will inspect natural generalizations of this construction to settings in which additional convexity structure is involved. The talk is based on joint work with N. Laustsen, M. Taylor, P. Tradacete and V. Troitsky.

The Steiner Tree Problem is a classical problem in combinatorial optimization. Given a set of vertices of a graph, it requires to connect such set with a minimum-weight subgraph. We consider this problem after restricting the input to some class of graphs characterized by a small set of forbidden induced subgraphs. We show a complexity dichotomy for the edge-weighted version of this problem restricted to (H_1,H_2)-free graphs and a complexity dichotomy for the vertex-weighted version restricted to H-free graphs. The rst conclusion that can be drawn from our work is that there exists an innite family of graphs H such that Vertex Steiner Tree can be solved in polynomial time for H-free graphs, whereas there exist only two graphs H for which this holds for Edge Steiner Tree. The second conclusion is that Edge Steiner Tree can be solved in polynomial time for (H_1,H_2)-free graphs if and only if the treewidth of the class of (H_1,H_2)-free graphs is bounded (subject to P≠NP). The talk is based on joint work with: Nick Brettel, Matthew Johnson and Daniël Paulusma (Durham University), Hans Bodlaender and Erik Jan van Leeuwen (Utrecht University).