We present some dimensionality reduction techniques for global optimization algorithms, in order to increase their scalability. Inspired by ideas in machine learning, and extending the approach of random projections in Zhang et al (2016), we present some new algorithmic approaches for global optimisation with theoretical guarantees of good behaviour and encouraging numerical results.
 

Given a representation of Gal_{Q_p} with coefficients in a p-adically complete local ring R, Fukaya and Kato have conjectured the existence of a canonical trivialization of the determinant of a certain cohomology complex.  When R=Z_p and the representation is a lattice in a de Rham representation, this trivialization should be related to the \varepsilon-factor of the corresponding Weil--Deligne representation.  Such a trivialization has been constructed for certain crystalline Galois representations, by the work of a number of authors. I will explain how to extend these trivializations to certain families of crystalline Galois representations.  This is joint work with Otmar Venjakob.

In this talk, I will give a brief overview of the Langlands program and Langlands functoriality with reference to the examples of Galois representations attached to cusp forms and the Jacquet-Langlands correspondence for $\mathrm{GL}_2$. I will then explain how one can generalise this idea, sketching a proof of a Jacquet-Langlands type correspondence from $\mathrm{U}_n(B)$, where $B$ is a quaternion algebra, to $\mathrm{Sp}_{2n}$ and showing that one can attach Galois representations to regular algebraic cuspidal automorphic representations of $\mathrm{Sp}_{2n}$.
 

Dedalus is a new open-source framework for solving general partial differential equations using spectral methods.  It is designed for maximum extensibility and incorporates features such as symbolic equation entry, custom domain construction, and automatic MPI parallelization.  I will briefly describe key algorithmic features of the code, including our sparse formulation and support for general tensor calculus in curvilinear domains.  I will then show examples of the code’s capabilities with various applications to astrophysical and geophysical fluid dynamics, including a compressible flow benchmark against a finite volume code, and direct numerical simulations of turbulent glacial melting

In my research I model three components of the Earth system: the ice sheets, the ocean, and the solid Earth. In the first half of this talk I will describe the traditional approach that is used to model the impact of ice sheet growth and decay on global sea-level change and solid Earth deformation. I will then go on to explain how collaboration across the fields of glaciology, geodynamics and seismology is providing exciting new insight into feedbacks between ice dynamics and solid Earth deformation.

In July 2011, the observation of a massive phytoplankton bloom underneath a sea ice–covered region of the Chukchi Sea shifted the scientific consensus that regions of the Arctic Ocean covered by sea ice were inhospitable to photosynthetic life. Although the impact of widespread phytoplankton blooms under sea ice on Arctic Ocean ecology and carbon fixation is potentially marked, the prevalence of these events in the modern Arctic and in the recent past is, to date, unknown. We investigate the timing, frequency, and evolution of these events over the past 30 years. Although sea ice strongly attenuates solar radiation, it has thinned significantly over the past 30 years. The thinner summertime Arctic sea ice is increasingly covered in melt ponds, which permit more light penetration than bare or snow-covered ice. We develop a simple mathematical model to investigate these physical mechanisms. Our model results indicate that the recent thinning of Arctic sea ice is the main cause of a marked increase in the prevalence of light conditions conducive to sub-ice blooms. We find that as little as 20 years ago, the conditions required for sub-ice blooms may have been uncommon, but their frequency has increased to the point that nearly 30% of the ice-covered Arctic Ocean in July permits sub-ice blooms. Recent climate change may have markedly altered the ecology of the Arctic Ocean.

We determine the hydrodynamic limit of some kinetic equations with either stochastic Vlasov force term or stochastic collision kernel. We obtain stochastic second-order parabolic equations at the limit. In the regime we consider, we also observe (or do not observe) some phenomena of enhanced diffusion. Joint works with Nils Caillerie, Arnaud Debussche, Martina Hofmanová.