University of Nottingham

The geometric Langlands correspondence relates rank n integrable connections on a complex Riemann surface $X$ to regular holonomic D-modules on  $Bun_n(X)$, the moduli stack of rank n vector bundles on $X$.  If we replace $X$ by a smooth irreducible curve over a finite field of characteristic p then there is a version of the geometric Langlands correspondence involving $l$-adic perverse sheaves for $l\neq p$.  In this lecture we consider the case $l=p$, proposing a $p$-adic version of the geometric Langlands correspondence relating convergent $F$-isocrystals on $X$ to arithmetic $D$-modules on $Bun_n(X)$.

In this talk we present an overview of some recent developments concerning the a posteriori error analysis and adaptive mesh design of $h$- and $hp$-version discontinuous Galerkin finite element methods for the numerical approximation of second-order quasilinear elliptic boundary value problems. In particular, we consider the derivation of computable bounds on the error measured in terms of an appropriate (mesh-dependent) energy norm in the case when a two-grid approximation is employed. In this setting, the fully nonlinear problem is first computed on a coarse finite element space $V_{H,P}$. The resulting 'coarse' numerical solution is then exploited to provide the necessary data needed to linearise the underlying discretization on the finer space $V_{h,p}$; thereby, only a linear system of equations is solved on the richer space $V_{h,p}$. Here, an adaptive $hp$-refinement algorithm is proposed which automatically selects the local mesh size and local polynomial degrees on both the coarse and fine spaces $V_{H,P}$ and $V_{h,p}$, respectively. Numerical experiments confirming the reliability and efficiency of the proposed mesh refinement algorithm are presented.

I will discuss some of new concepts and objects of two-dimensional number theory: 

how the same object can be studied via low dimensional noncommutative theories or higher dimensional commutative ones, 

what is higher Haar measure and harmonic analysis and how they can be used in representation theory of non locally compact groups such as loop groups and Kac-Moody groups, 

how classical notions split into two different notions on surfaces on the example of adelic structures, 

what is the analogue of the double quotient of adeles on surfaces and how one

could approach automorphic functions in geometric dimension two.

Stiff systems of ODEs arise commonly when solving PDEs by spectral methods,

so conventional explicit time-stepping methods require very small time steps.

The stiffness arises predominantly through the linear terms, and these

terms can be handled implicitly or exactly, permitting larger time steps.

This work develops and investigates a class of methods known as

'exponential time differencing'. These methods are shown to have a

number of advantages over the more well-known linearly implicit

methods and integrating factor methods.

The formation of steady patterns in one space dimension is generically

governed, at small amplitude, by the Ginzburg-Landau equation.

But in systems with a conserved quantity, there is a large-scale neutral

mode that must be included in the asymptotic analysis for pattern

formation near onset. The usual Ginzburg-Landau equation for the amplitude

of the pattern is then coupled to an equation for the large-scale mode.

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These amplitude equations show that for certain parameters all regular

periodic patterns are unstable. Beyond the stability boundary, there

exist stable stationary solutions in the form of spatially modulated

patterns or localised patterns. Many more exotic localised states are

found for patterns in two dimensions.

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Applications of the theory include convection in a magnetic field,

providing an understanding of localised states seen in numerical

simulations.

The mechanisms for the selection of the propagation speed of waves

connecting unstable to stable states will be discussed in the

spatially non-homogeneous case, the differences from the very

well-studied homogeneous version being emphasised.

This seminar will be held at the Rutherford Appleton Laboratory near Didcot.

Abstract:

Numerical calculations of laminar flow in a two-dimensional channel with a sudden expansion exhibit a symmetry-breaking bifurcation at Reynolds number 40.45 when the expansion ratio is 3:1. In the experiments reported by Fearn, Mullin and Cliffe [1] there is a large perturbation to this bifurcation and the agreement with the numerical calculations is surprisingly poor. Possible reasons for this discrepancy are explored using modern techniques for uncertainty quantification.

When experimental equipment is constructed there are, inevitably, small manufacturing imperfections that can break the symmetry in the apparatus. In this work we considered a simple model for these imperfections. It was assumed that the inlet section of the channel was displaced by a small amount and that the centre line of the inlet section was not parallel to the centre line of the outlet section. Both imperfections were modelled as normal random variables with variance equal to the manufacturing tolerance. Thus the problem to be solved is the Navier-Stokes equations in a geometry with small random perturbations. A co-ordinate transformation technique was used to transform the problem to a fixed deterministic domain but with random coefficient appearing in the transformed Navier-Stokes equations. The resulting equations were solved using a stochastic collocation technique that took into account the fact that the problem has a discontinuity in parameter space arising from the bifurcation structure in the problem.

The numerical results are in the form of an approximation to a probability measure on the set of bifurcation diagrams. The experimental data of Fearn, Mullin and Cliffe are consistent with the computed solutions, so it appears that a satisfactory explanation for the large perturbation can be provided by manufacturing imperfections in the experimental apparatus.

The work demonstrates that modern methods for uncertainty quantification can be applied successfully to a bifurcation problem arising in fluid mechanics. It should be possible to apply similar techniques to a wide range of bifurcation problems in fluid mechanics in the future.

References:

[1] R M Fearn, T Mullin and K A Cliffe Nonlinear flow phenomena in a symmetric sudden expansion, J. Fluid Mech. 211, 595-608, 1990.