Imperial College, London

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.

We describe discretisations of the shallow water equations on

the sphere using the framework of finite element exterior calculus. The

formulation can be viewed as an extension of the classical staggered

C-grid energy-enstrophy conserving and

energy-conserving/enstrophy-dissipating schemes which were defined on

latitude-longitude grids. This work is motivated by the need to use

pseudo-uniform grids on the sphere (such as an icosahedral grid or a

cube grid) in order to achieve good scaling on massively parallel

computers, and forms part of the multi-institutional UK “Gung Ho”

project which aims to design a next generation dynamical core for the

Met Office Unified Model climate and weather prediction system. The

rotating shallow water equations are a single layer model that is

used to benchmark the horizontal component of numerical schemes for

weather prediction models.

We show, within the finite element exterior calculus framework, that it

is possible

to build numerical schemes with horizontal velocity and layer depth that

have a con-

served diagnostic potential vorticity field, by making use of the

geometric properties of the scheme. The schemes also conserve energy and

enstrophy, which arise naturally as conserved quantities out of a

Poisson bracket formulation. We show that it is possible to modify the

discretisation, motivated by physical considerations, so that enstrophy

is dissipated, either by using the Anticipated Potential Vorticity

Method, or by inducing stabilised advection schemes for potential

vorticity such as SUPG or higher-order Taylor-Galerkin schemes. We

illustrate our results with convergence tests and numerical experiments

obtained from a FEniCS implementation on the sphere.

In this talk, we first study the Mean Curvature Flow, an evolution equation for submanifolds of some Euclidean space. We review a famous monotonicity formula of Huisken and its application to classifying so-called Type I singularities. Then, we discuss the Ricci Flow, which might be seen as the intrinsic analog of the Mean Curvature Flow for abstract Riemannian manifolds. We explain how Huisken's classification of Type I singularities can be adopted to this intrinsic setting, using monotone quantities found by Perelman.

Lusztig discover an integral lift of the Frobenius morphism for algebraic groups in positive characteristic to quantum groups at a root of unity. We will describe how this map may be constructed via the Hall algebra realization of a quantum group.

We shall begin with simple Weyl type asymptotic formulae for the spectrum of Dirichlet Laplacians and eventually prove a new result which I have recently obtained, jointly with J. Dolbeault and M. Loss. Following Eden and Foias, we derive a matrix version of a generalised Sobolev inequality in one dimension. This allows us to improve on the known estimates of best constants in Lieb-Thirring inequalities for the sum of the negative eigenvalues for multi-dimensional Schrödinger operators.

Bio: Ari Laptev received his PhD in Mathematics from Leningrad University (LU) in 1978, under the supervision of Michael Solomyak. He is well known for his contributions to the Spectral Theory of Differential Operators. Between 1972 - 77 and 1977- 82 he was employed as a junior researcher and as Assistant Professor at the Mathematics & Mechanics Department of LU. In 1981- 82 he held a post-doc position at the University of Stockholm and in 1982 he lost his position at LU due to his marriage to a British subject. Up until his emigration to England in 1987 he was working as a builder, constructing houses in small villages in the Novgorod district of Russia. In 1987 he was employed in Sweden, first as a lecturer at Linköping University and then from 1992 at the Royal Institute of Technology (KTH). In 1999 he became a professor at KTH and also Vice Chairman of its Mathematics Department. In 1992 he was granted Swedish citizenship. Ari Laptev was the President of the Swedish Mathematical Society from 2001 to 2003 and the President of the Organizing Committee of the Fourth European Congress of Mathematics in Stockholm in 2004. From January 2007 he has been employed by Imperial College London. Ari Laptev has supervised twelve PhD students. From January 2007 until the end of 2010 he is President of the European Mathematical Society.