Virtual

C*-algebras associated to etale groupoids appear as a versatile construction in many contexts. For instance, groupoid C*-algebras allow for implementation of natural one-parameter groups of automorphisms obtained from continuous cocycles. This provides a path to quantum statistical mechanical systems, where one studies equilibrium states and ground states. The early characterisations of ground states and equilibrium states for groupoid C*-algebras due to Renault have seen remarkable refinements. It is possible to characterise in great generality all ground states of etale groupoid C*-algebras in terms of a boundary groupoid of the cocycle (joint work with Laca and Neshveyev). The steps in the proof employ important constructions for groupoid C*-algebras due to Renault.

I will talk about the joint moments of characteristic polynomials of random unitary matrices and their derivatives. In joint work with Jon Keating and Jon Warren we establish the asymptotics of these quantities for general real values of the exponents as the size N of the matrix goes to infinity. This proves a conjecture of Hughes from 2001. In subsequent joint work with Benjamin Bedert, Mustafa Alper Gunes and Arun Soor we focus on the leading order coefficient in the asymptotics, we connect this to Painleve equations for general values of the exponents and obtain explicit expressions corresponding to the so-called classical solutions of these equations.

I will discuss an analogue of the CHY formalism in AdS. Considering the biadjoint scalar theory on AdS, I will explain how to rewrite all the tree-level amplitudes as an integral over the moduli space of punctured Riemann spheres. Contrary to the flat space, the scattering equations are operator-valued. The resulting formula is motivated via a bosonic ambitwistor string on AdS and can be proven to be equivalent to the corresponding Witten diagram computation by applying a series of contour deformations.

 I will review what is known and not known about the joint moments of the characteristic polynomials of random unitary matrices and their derivatives. I will then explain some recent results which relate the joint moments to an interesting class of measures, known as Hua-Pickrell measures. This leads to the proof of a conjecture, due to Chris Hughes in 2000, concerning the asymptotics of the joint moments, as well as establishing a connection between the measures in question and one of the Painlevé equations.

The session will be a panel discussion addressing practical aspects of doing a research degree. We will take questions from the audience so will discuss whatever people wish to ask us, but we expect to talk about the process of applying, why you might want to consider doing a research degree, the experience of doing research, and what people do after they have completed their degree.

In this interactive workshop, we'll discuss what mathematicians are looking for in written solutions.  How can you set out your ideas clearly, and what are the standard mathematical conventions?

This session is likely to be most relevant for first-year undergraduates, but all are welcome.

This session is particularly aimed at fourth-year and OMMS students who are completing a dissertation this year. The talk will be given by Dr Richard Earl who chairs Projects Committee. For many of you this will be the first time you have written such an extended piece on mathematics. The talk will include advice on planning a timetable, managing the workload, presenting mathematics, structuring the dissertation and creating a narrative, providing references and avoiding plagiarism.

Steklov eigenproblems and their variants (where the spectral parameter appears in the boundary condition) arise in a range of useful applications. For instance, understanding some properties of the mixed Steklov-Neumann eigenfunctions tells us why we shouldn't use coffee cups for expensive brandy. 

In this talk I'll present a high-accuracy discretization strategy for computing Steklov eigenpairs. The strategy can be used to study questions in spectral geometry, spectral optimization and to the solution of elliptic boundary value problems with Robin boundary conditions.

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A link for the talk will be sent to our mailing list a day or two in advance.  If you are not on the list and wish to be sent a link, please contact @email.

Averages of multiplicative eigenvalue statistics of Haar distributed unitary matrices are Toeplitz determinants, and asymptotics for these determinants are now well understood for large classes of symbols, including symbols with gaps and (merging) Fisher-Hartwig singularities. Similar averages for Haar distributed orthogonal matrices are Toeplitz+Hankel determinants. Some asymptotic results for these determinants are known, but not in the same generality as for Toeplitz determinants. I will explain how one can systematically deduce asymptotics for averages in the orthogonal group from those in the unitary group, using a transformation formula and asymptotics for certain orthogonal polynomials on the unit circle, and I will show that this procedure leads to asymptotic results for symbols with gaps or (merging) Fisher-Hartwig singularities. The talk will be based on joint work with Gabriel Glesner, Alexander Minakov and Meng Yang.