Mathematical Colloquium

Commutative algebraic groups are geometric objects defined by equations which are equipped with the additional structure of an Abelian group and therefore are algebraic analogues of commutative Lie groups. Because they are described by equations, one can ask both geometric and arithmetic questions about them, and the interplay between these aspects is one of the main areas of research in this field. I shall report on recent joint work with T. Suzuki in which we establish a finiteness property of the group of rational points for smooth connected "unipotent" algebraic groups, answering a question posed by Oesterlé. This result follows from a conjecture of Bosch-Lütkebohmert-Raynaud about the existence of Néron models of smooth algebraic groups, settling which is the main result of our joint work. This talk will be accessible to a general mathematical audience.

In 1911, Otto Toeplitz posed the following question: Does every closed Jordan curve have an inscribed square? This question remains unanswered. We will discuss several variations of this question, focusing in particular on the above question for smooth closed Jordan curves, which has a positive answer.

We establish necessary and sufficient conditions for stochastic invariance of closed subsets in Hilbert spaces for solutions to infinite-dimensional stochastic differential equations (SDEs) under mild assumptions on the coefficients. Our first characterization is formulated in terms of certain normal vectors to the invariance set and requires differentiability only of the dispersion operator, but not of the diffusion coefficient itself. The condition involves a suitable corrected drift expressed through the dispersion operator and its Moore-Penrose pseudoinverse, extending the classical Stratonovich correction term to the present low-regularity setting. Our second characterization is given in terms of the positive maximum principle for the infinitesimal generator of the associated diffusion process. We illustrate our characterizations in the case of invariant manifolds.
This presentation is based on joint work with Eduardo Abi Jaber (École Polytechnique, Paris, France).

We consider numerical methods for the dynamical core in numerical weather prediction (NWP). The current trend in NWP is to increase the resolution, making high order Discontinuous Galerkin (DG) methods competitive. These are well suited for High Performance Computing due to their high arithmetic intensity. The overall design goal is thus a suitable highly parallel stable and computationally efficient DG implementation.
The talk gives an overview for non specialists of dynamical cores in NWP and presents results of a collaboration between Lund University and Environment and Climate Change Canada (ECCC) on a current redesign of their dynamical core along these lines within the code WxFactory.
The PDEs of interest are the Euler equations in 3D at low Mach numbers with a gravity source term. To deal with the gravitational source term, a well-balanced method is needed, suitable for large domains. Additionally, the scheme requires a numerical flux and time integration suitable for low Mach numbers, and some form of nonlinear stability such as entropy stability, suitable for large simulation times.
We use a well-balanced Direct Flux Reconstruction implementation on GL nodes on quadrilateral meshes, a variant of the AUSM+up flux, and an exponential integrator in time. To achieve entropy stability, we are in the process of incorporating an artificial diffusion term and a relaxation technique to stabilize the time integration. We provide numerical results showing the performance on atmospheric test cases in 2D and 3D.
This is joint work with Stéphane Gaudreault, Shoyon Panday, Carlos Pereira Frontado (ECCC), and Herman Efraimsson, Martin Hindle, Kateryna Tymoshchuk (LU).

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