In this talk we consider the issue of mass loss in fragmentation models due to 'shattering'. As a solution we propose a hybrid discrete/continuous model whereby the smaller particles are considered as having discrete mass, whilst above a certain cut-off, mass is taken to be a continuous variable. The talk covers the development of such a model, its initial analysis via the theory of operator semigroups and its numerical approximation using a finite volume discretisation.

We explore the use of applying multiple preconditioners for solving linear systems arising in simulations of incompressible two-phase flow. In particular, we use a selective MPGMRES algorithm, for which the search space grows linearly throughout the iterative solver, and block preconditioners based on Schur complement approximations

We introduce a discontinuous Galerkin finite element method (DGFEM) for Hamilton–Jacobi–Bellman equations on piecewise curved domains, and prove that the method is consistent, stable, and produces optimal convergence rates. Upon utilising a long standing result due to N. Krylov, we may characterise the Monge–Ampère equation as a HJB equation; in two dimensions, this HJB equation can be characterised further as uniformly elliptic HJB equation, allowing for the application of the DGFEM

We propose a new parameter estimation technique for SDEs, based on the inverse problem of finding a forward operator describing the evolution of temporal data. Nonlinear dynamical systems on a state-space can be lifted to linear dynamical systems on spaces of higher, often infinite, dimension. Recently, much work has gone into approximating these higher-dimensional systems with linear operators calculated from data, using what is called Dynamic Mode Decomposition (DMD). For SDEs, this linear system is given by a second-order differential operator, which we can quickly calculate and compare to the DMD operator.

In 2016-17, Fjordholm, Kappeli, Mishra and Tadmor developed a numerical method by which one could compute measure-valued solutions to systems of hyperbolic conservation laws with either measure-valued or deterministic initial data. In this talk I will discuss the ideas behind this method, and discuss how it can be adapted to systems of quasi-linear parabolic PDEs whose nonlinearity fails to satisfy a monotonicity condition.

The basic mathematical models that describe the behavior of fluid flows date back to the eighteenth century, and yet many phenomena observed in experiments are far from being well understood from a theoretical viewpoint. For instance, especially challenging is the study of fundamental stability mechanisms when weak dissipative forces (generated, for example, by molecular friction) interact with advection processes, such as mixing and stirring. The goal of this talk is to have an overview on recent results on a variety of aspects related to hydrodynamic stability, such as the stability of vortices and laminar flows, the enhancement of dissipative force via mixing, and the statistical description of turbulent flows.

Serre's uniformity question concerns the possible ways the Galois group of Q can act on the p-torsion of an elliptic curve over Q. In this talk I will survey what is known about this question, and describe two recent results related to the Chabauty-Kim method. The first, which is joint work with Jennifer Balakrishnan, Steffen Muller, Jan Tuitman and Jan Vonk, completes the classification of elliptic curves over Q with split Cartan level structure. The second, which is work in progress with Samuel Le Fourn, Samir Siksek and Jan Vonk, concerns the applicability of the Chabauty-Kim method in determining the elliptic curves with non-split Cartan level structure.
 

In this talk, I will present the construction of a family of solutions to
vacuum Einstein equations which consist of an arbitrary number of high
frequency waves travelling in different directions. In the high frequency
limit, our family of solutions converges to a solution of Einstein equations
coupled to null dusts. This construction is an illustration of the so called
backreaction, studied by physicists (Isaacson, Burnet, Green, Wald...) : the
small scale inhomogeneities have an effect on the large scale dynamics in
the form of an energy impulsion tensor in the right-hand side of Einstein
equations. This is a joint work with Jonathan Luk (Stanford).

Some of the most studied examples of conformal field theories
include
the Wess-Zumino-Witten models. These are conformal field theories exhibiting
affine Lie algebra symmetry at non-negative integers levels. In this talk I
will
discuss conformal field theories exhibiting affine Lie algebra symmetry at
certain rational (hence fractional) levels whose structure is arguably even
more intricate than the structure of the non-negative integer levels,
provided
one is prepared to look beyond highest weight modules.

Hardy's axiomatic approach to quantum theory revealed that just one axiom
distinguishes quantum theory from classical probability theory: there should
be continuous reversible transformations between any pair of pure states. It
is the single word `continuous' that gives rise to quantum theory. This
raises the question: Does there exist a finite theory of quantum physics
(FTQP) which can replicate the tested predictions of quantum theory to
experimental accuracy? Here we show that an FTQP based on complex Hilbert
vectors with rational squared amplitudes and rational phase angles is
possible providing the metric of state space is based on p-adic rather than
Euclidean distance. A key number-theoretic result that accounts for the
Uncertainty Principle in this FTQP is the general incommensurateness between
rational $\phi$ and rational $\cos \phi$. As such, what is often referred to
as quantum `weirdness' is simply a manifestation of such number-theoretic
incommensurateness. By contrast, we mostly perceive the world as classical
because such incommensurateness plays no role in day-to-day physics, and
hence we can treat $\phi$ (and hence $\cos \phi$) as if it were a continuum
variable. As such, in this FTQP there are two incommensurate Schr\"{o}dinger
equations based on the rational differential calculus: one for rational
$\phi$ and one for rational $\cos \phi$. Each of these individually has a

simple probabilistic interpretation - it is their merger into one equation
on the complex continuum that has led to such problems over the years. Based
on this splitting of the Schr\"{o}dinger equation, the measurement problem
is trivially solved in terms of a nonlinear clustering of states on $I_U$.
Overall these results suggest we should consider the universe as a causal
deterministic system evolving on a finite fractal-like invariant set $I_U$
in state space, and that the laws of physics in space-time derive from the
geometry of $I_U$. It is claimed that such a  deterministic causal FTQP will
be much easier to synthesise with general relativity theory than is quantum
theory.