I will talk about recent work with Jack Thorne in which we find the average size of the Selmer group for a family of genus-2 curves by analyzing a graded Lie algebra of type E_8. I will focus on the role representation theory plays in our proofs.

When solving partial differential equations driven by additive spatial white noise, the efficient sampling of white noise realizations can be challenging. In this talk we focus on the efficient sampling of white noise using quasi-random points in a finite element method and multilevel Quasi Monte Carlo (MLQMC) setting. This work is an extension of previous research on white noise sampling for MLMC.

We express white noise as a wavelet series expansion that we divide in two parts. The first part is sampled using quasi-random points and contains a finite number of terms in order of decaying importance to ensure good QMC convergence. The second part is a correction term which is sampled using standard pseudo-random numbers.

We show how the sampling of both terms can be performed in linear time and memory complexity in the number of mesh cells via a supermesh construction. Furthermore, our technique can be used to enforce the MLQMC coupling even in the case of non-nested mesh hierarchies. We demonstrate the efficacy of our method with numerical experiments.

With growing population of humans being clustered in large cities and connected by fast routes more suitable environments for epidemics are being created. Topped by rapid mutation rate of viral and bacterial strains, epidemiological studies stay a relevant topic at all times. From the beginning of 2019, the World Health Organization publishes at least five disease outbreak news including Ebola virus disease, dengue fever and drug resistant gonococcal infection, the latter is registered in the United Kingdom.

To control the outbreaks it is necessary to gain information on mechanisms of appearance and evolution of pathogens. Close to all disease-causing virus and bacteria undergo a specialization towards a human host from the closest livestock or wild fauna of a shared habitat. Every strain (or subtype) of a pathogen has a set of characteristics (e.g. infection rate and burst size) responsible for its success in a new environment, a host cell in case of a virus, and with the right amount of skepticism that set can be framed as fitness of the pathogen. In our model, we consider a population of a mutating strain of a virus. The strain specialized towards a new host usually remains in the environment and does not switch until conditions get volatile. Two subtypes, wild and mutant, of the virus share a host. This talk will illustrate findings on an explicitly independent cycling coexistence of the two subtypes of the parasite population. A rare transcritical bifurcation of limit cycles is discussed. Moreover, we will find conditions when one of the strains can outnumber and eventually eliminate the other strain focusing on an infection rate as fitness of strains.

In this talk, we will consider how two very different atmospheric phenomena, the terrestrial tropical cyclone and the martian polar vortex, can be described within a single simplified dynamical framework based on the forced shallow water equations. Dynamical forcings include angular momentum transport by secondary (transverse) circulations and local heating due to latent heat release. The forcings act in very different ways in the two systems but in both cases lead to distinct annular distributions of potential vorticity, with a local vorticity maximum at a finite radius surrounding a central minimum.  In both systems, the resulting vorticity distributions are subject to shear instability and the degree of eddy growth versus annular persistence can be examined explicitly under different forcing scenarios.

Inertia-gravity waves (IGWs) are ubiquitous in the ocean and the atmosphere. Once generated (by tides, topography, convection and other processes), they propagate and scatter in the large-scale, geostrophically-balanced background flow. I will discuss models of this scattering which represent the background flow as a random field with known statistics. Without assumption of spatial scale separation between waves and flow, the scattering is described by a kinetic equation involving a scattering cross section determined by the energy spectrum of the flow. In the limit of small-scale waves, this equation reduces to a diffusion equation in wavenumber space. This predicts, in particular, IGW energy spectra scaling as k^{-2}, consistent with observations in the atmosphere and ocean, lending some support to recent claims that (sub)mesoscale spectra can be attributed to almost linear IGWs.  The theoretical predictions are checked against numerical simulations of the three-dimensional Boussinesq equations.
(Joint work with Miles Savva and Hossein Kafiabad.)