To welcome those new to the Postgraduate Forum, we begin the year with a special session comprising of short talks from various speakers:

Nik Alexandrakis - The Kernel of Weyl-Dirac Operators

Zachary Hall - Fusion systems of groups

Ai Guan - Deformation theory

Frankie Higgs - Scaling limits of random growth processes

Joe Wall - What is rigidity theory?

Max Arnott - Kernels of bounded operators on transfinite Banach sequence spaces

If someone gives you a variety with a singular point, you can try and get some understanding of what the singularity looks like by taking its "link", that is you take the boundary of a neighbourhood of the singular point. For example, the link of the complex plane curve with a cusp y^2 = x^3 is a trefoil knot in the 3-sphere. I want to talk about the links of a class of 3-fold singularities which come up in Mori theory: the compound Du Val (cDV) singularities. These links are 5-dimensional manifolds. It turns out that many cDV singularities have the same 5-manifold as their link, and to tell them apart you need to keep track of some extra structure (a contact structure). In joint work with Y. Lekili, we use symplectic cohomology to distinguish the contact structures on many these links.

We will describe the class of bar-joint rigid frameworks which are restricted to move on the surface of a cylinder, where if there is a point at p, there is a symmetric point at -p. We will introduce the base graphs for this class, and give rigidity preserving moves which enable us to generate the entire class of graphs.

In this talk the objects of interest are random interface models: Gaussian fields that model random separating surfaces between different phases of matter. Among them, semiflexible polymers describe surfaces that tend to preserve either the mean height or the mean curvature. This is translated by saying that their energy is governed by a mixed gradient and Laplacian interaction. Among them are the discrete Gaussian free field and the discrete membrane model. The strength of the two operators is governed by two parameters, called lateral tension and bending rigidity, which may depend on the size of the polymer. In this work we show a phase transition in the large-size limit according to the magnitude of the tension and the rigidity: we prove that the scaling limit is, respectively, a Gaussian free field, a “mixed” random distribution and the continuum membrane, or Bilaplacian, model in three different regimes.

A framework is a combination of an graph and a realisation of the vertices of this graph in some space. If the vertices of a graph are realised in d-dimensional Euclidean space then the rigidity of frameworks is well understood for d = 1, 2, but the situation is less clear for higher dimensions. When d=3, a natural consideration is to restrict the realisation to some surface. We give an overview of the historical results in low-dimensional Euclidean space, and a similar description for particular surfaces, before giving an insight into ongoing work that looks to characterise rigid frameworks realised on two non-concentric spheres.

I'll discuss the following problem posed by Feliks Przytycki. Let (X,mu) be a Polish space with a Borel probability measure and let T:X->X be a Borel map preserving mu. Let Y be a Polish space and S:Y->Y, p:X->Y be Borel maps such that pS=Tp. Does mu lift to a Borel probability nu on Y such that S preserves nu?

As is rather well-known, an m by n rectangular grid of squares (or rectangles)  becomes rigid if diagonal braces are added in a particular combinatorial way. The "braces subgraph" of K_{m,n} must be connected and spanning (Bolker-Crapo, 1977). We consider braced grids for non-Euclidean norms and obtain analogous results. The combinatorial setting is richer however since both diagonal braces are required to rigidify a 1 by 1 grid, and the appropriate braces graph is a subgraph of the bicoloured multigraph K_{m,n}^2.

The aim of the next two Postgraduate forums is to introduce the methods commonly used to study a class of random growth processes, in which at each step particles are attached to the boundary of a cluster. In the first lecture we will start by providing examples of the real-world processes that we would like to understand, before describing in detail how models are built in order to study these processes. We will define the Hastings–Levitov model, which is a one-parameter family of models formed using conformal maps that is used to describe Laplacian growth. Finally, we will introduce the different approaches commonly used to find a scaling limit for these processes and consider the phase transitions in the scaling limit of the Hastings–Levitov HL(alpha) model as the parameter alpha varies.

Imagine a plane tree together with a configuration of particles (cars) at each vertex.Each car tries to park on its node, and if the latter is occupied, it moves downward towards the root trying to find an empty slot.When the underlying plane tree is a critical Galton--Watson conditioned to be large, and when the cars arrivals are i.i.d. on each vertex, we observe a phase transition:- when the density of cars is small enough, all but a few manage to park safely,- whereas when the density of cars is high enough, a positive fraction of them do not manage to park and exit through the root of the tree.The critical density is an explicit function of the first two moments of the offspring distribution and cars arrivals (C. & Hénard 2019).We shall give a new point of view on this process by coupling it with the ubiquitous Erdös--Rényi random graph process.This enables us to fully understand the (dynamical) phase transition in the scaling limit by relating it to the multiplicative coalescent process. The talk is based on a joint work with Olivier Hénard and an ongoing project with Alice Contat.

Last week, George introduced the notion of a conformal random growth model: a mathematical model of a real-world growth process constructed by composing conformal maps. This time we will look at some details of these models and some major results, paying particular attention to how the complex-analytic setting gives us access to useful techniques that are not available for models defined on a discrete lattice.

This talk will outline the different approaches used when discussing different burnside rings, and the general idea behind what we mean by a burnside ring

Bukh and Zhou conjectured that the expectation of the length of the longest common subsequence of two i.i.d random permutations of size n is greater than  √n. This problem is related to the Ulam-Hammersley problem. We recall the classical results for the uniform case as well as partial answers for the conjugation invariant case.

In this talk we will see a brief overview of the basic concepts of representation theory of algebras. We will introduce the notions of quiver, representation, (bound) path algebra and the equivalence between modules and representations. Then we will also provide some examples of algebras for which it is possible to know all the indecomposable finitely generated modules up to isomorphism.

This talk will be an investigation into the class of inversion symmetric bar-joint rigid frameworks, restricted to move on a cylinder. We show how our class of graphs can be generated from two base graphs and rigidity preserving moves.

Roughly speaking, a group with Kazhdan's property (T) presents a dichotomy. When acting on a Hilbert space by isometries, it either fixes vectors or moves every vector by at least a fixed quantity. Checking this definition, however, requires making a statement about every unitary representation of the group. In this introductory talk we analyse the implications of property (T) on the spectra of a selected collection of operators. The ultimate goal is to understand that property (T) can be characterized by a spectral condition on a remarkable element of the group ring.

We study the coefficients of the characteristic polynomial (also called secular coefficients) of random unitary matrices drawn from the Circular Beta Ensemble (i.e. the joint probability density of the eigenvalues is proportional to the product of the power beta of the mutual distances between the points). We study the behavior of the secular coefficients when the degree of the coefficient and the dimension of the matrix tend to infinity. The order of magnitude of this coefficient depends on the value of the parameter beta, in particular, for beta = 2, we show that the middle coefficient of the characteristic polynomial of the Circular Unitary Ensemble converges to zero in probability when the dimension goes to infinity, which solves an open problem of Diaconis and Gamburd. We also find a limiting distribution for some renormalized coefficients in the case where beta > 4. In order to prove our results, we introduce a holomorphic version of the Gaussian Multiplicative Chaos, and we also make a connection with random permutations following the Ewens measure.