University College Dublin

The study of periods of automorphic forms is a key theme in the Langlands program and has become an important tool to tackle various problems in Number Theory and Arithmetic Geometry.  For instance, Waldspurger formula and its generalisations have created a fertile ground for numerous arithmetic applications. In recent years, the conjectures of Sakellaridis and Venkatesh (and then Ben-Zvi, Sakellaridis, and Venkatesh) in the context of spherical varieties has led to a deeper understanding of automorphic periods and their relation to special values of $L$-functions. In this talk, I present work in progress aimed at looking at certain non-spherical cases. Precisely, I will describe a new integral representation of the degree 12 "exterior square x standard" $L$-function on generic cusp forms on $\mathrm{GU}(2,2) \times \mathrm{GL}(2)$ (or $\mathrm{GL}(4) \times \mathrm{GL}(2)$) and how it can be used to relate the non-vanishing of its central value to a certain cohomological period.  If time permits, I will describe how the same strategy applies to the case of $\mathrm{GSp}(6) \times \mathrm{GL}(2)$. This is joint work with Armando Gutierrez Terradillos.

The Wilson–Cowan population model of neural activity has greatly influenced our understanding of the mechanisms for the generation of brain rhythms and the emergence of structured brain activity. As well as the many insights that have been obtained from its mathematical analysis, it is now widely used in the computational neuroscience community for building large scale in silico brain networks that can incorporate the increasing amount of knowledge from the Human Connectome Project. In this talk, I will introduce a new neural population model in the spirit of that originally developed by Wilson and Cowan, albeit with the added advantage that it can account for the phenomena of event related synchronisation and de-synchronisation. This derived mean field model provides a dynamic description for the evolution of synchrony, as measured by the Kuramoto order parameter, in a large population of quadratic integrate-and-fire model neurons. As in the original Wilson–Cowan framework, the population firing rate is at the heart of our new model; however, in a significant departure from the sigmoidal firing rate function approach, the population firing rate is now obtained as a real-valued function of the complex valued population synchrony measure. To highlight the usefulness of this next generation Wilson–Cowan style model I will show how it can be deployed in a number of neurobiological contexts, providing understanding of the changes in power-spectra observed in EEG/MEG neuroimaging studies of motor-cortex during movement, insights into patterns of functional-connectivity observed during rest and their disruption by transcranial magnetic stimulation, and to describe wave propagation across cortex.

The study of 'moments' of random matrices (expectations of traces of powers of the matrix) is a rich and interesting subject, with fascinating connections to enumerative geometry, as discovered by Harer and Zagier in the 1980’s. I will give some background on this and then describe some recent work which offers some new perspectives (and new results). This talk is based on joint work with Fabio Deelan Cunden, Francesco Mezzadri and Nick Simm.