In this talk I will the discuss the concept of mollifying the moments of families of L-functions. I will begin with the Riemann zeta function as an illustrative example of mollified moments and their application to the zeros of L-functions. I will then give the relevant background on L-functions over function fields before discussing mollified moments in this setting and their application to non-vanishing results.

Instabilities are at the heart of several problem in fluid dynamics occurring in wide range of fields from engineering to geophysics to astrophysics. They are the onset points for the development of turbulence. Of particular interest in the current research problem is the magnetic Rayleigh Taylor instability (MRTI), where a lighter fluid accelerates into heavier fluid under the influence of gravity and in the presence of magnetic field. It is a ubiquitous phenomenon in astrophysics, occurring in a wide range of astrophysical systems from stars to supernova to accretion discs. While numerous studies have previously been carried out in hydrodynamic RTI, very little work was done in magnetic RTI. In the current research, we aim to study the fundamental problem using numerical simulations, particularly focusing on the evolution and late time mixing behaviour of fluids. The study uses a spectral code called Dedalus. The talk introduces audiences to the hydrodynamic instabilities, and the influence of fields on the growth of instability. Towards the end, preliminary results from the two-dimensional system are presented.

In probability theory, the Borel-Cantelli lemma can be used to determine the probability that infinitely many events occur in some sequence of events. In a dynamical setting, we may establish analogous properties on sequences of shrinking target sets. Typically, such target sets shrink to a point. In this talk, we shall focus on dynamical Borel-Cantelli properties in the context of sequences of sets which shrink to more general classes of sets.

Our work is based on Kolmogorov flow, first studied by Meshalkin & Sinai (1961), which is a sinusoidal velocity field with a two-dimensional, unidirectional profile, u= (0, sin x). This is maintained by an external force in a viscous fluid. It is known that this kind of fluid flow is unstable to large-scale jet motions, known as "zonostrophic instability", and this has recently been studied in a variety of settings both numerically and analytically. For example, Manfroi & Young (2002) have incorporated a beta-effect corresponding to a gradient of background planetary vorticity. Jet formation through instability can occur in the presence of the magnetic field and has implications observed in geophysical and astrophysical systems. Hughes et al. (2007).

In our study, we incorporate a mean magnetic field, which can be x-directed ("horizontal") or y-directed ("vertical") in our two-dimensional system and gives an MHD version of Kolmogorov flow. In a basic equilibrium state, magnetic field lines are straight for the case of vertical field and sinusoidal for horizontal field with an additional component of the external force balancing the resulting Lorentz force. As the basic state is independent of the y-coordinate we use Fourier analysis to study waves of wavenumber k in the y-direction, using the methods of classical stability theory and numerical solution of eigenvalue problems. We present some results using analytical approximations in the limit of k to 0, that is for large-scale jets. We also present some nonlinear results making use of the package Dedalus.

Nowcasting precipitation up to six hours ahead is a non-trivial problem for even modern-day numerical weather prediction systems (NWP) or ensemble weather forecasts. The parametrisation of our underlying hierarchical dynamic spatio-temporal model is motivated by a forward-time, centred-space finite difference solution to a collection of stochastic partial differential equations.

This research explores an approach which uses vector autoregressive model (VAR) in the problem of precipitation nowcasting based on the initial intensity and velocity. A new approach to radar nowcasting that we are exploring is the formulation of numerical solutions of the stochastic advection equation as a vector autoregressive (VAR) process with a sparse evolution operator.

My talk will be looking at the physical-dynamical system represented by the advection equation: it turns out there is a nice relationship between solving this SPDE and the vector autoregressive process which I will show.