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).

The ring $\mathbb{C}[[X]]$ of power series in one variable, over the complex field $\mathbb{C}$, is complete with respect to the $X$-adic absolute value. The elements of its field of fractions $\mathbb{C}((X))$ may also be viewed as the Laurent series over $\mathbb{C}$ with finite principal value (i.e. only finitely many terms $a_{i}X^{i}$ with $i\lt0$); moreover $\mathbb{C}[[X]]$ is the valuation ring of the $X$-adic valuation on $\mathbb{C}((X))$.
The first-order theory of $\mathbb{C}((X))$ (in a suitable language of valued fields) was axiomatized, and shown to be decidable, in the 60s by Ax and Kochen, and independently by Ershov: the key ingredient was “henselianity”, a first-order relic of completeness. In fact, henselianity together with axioms specifying the theory of the residue field $k$ of the valuation also suffices to axiomatize the theory of $k((X))$ for any $k$ of characteristic zero.
In mixed characteristic one analogon is the $p$-adic field $\mathbb{Q}_{p}$ which consists of formal sums over powers of a fixed prime number $p$ with coefficients from $\{0,\ldots,p-1\}$. In this setting too, henselianity is the key to unlocking an axiomatization of the complete theory. Similar axiomatizations and decidability results are known for any finitely ramified mixed characteristic henselian valued field.
Things are wildly different in positive characteristic: there is no known axiomatization of the complete theory of $\mathbb{F}_{p}((X))$, despite it being one of the most natural candidates. On the other hand, lots is known by now, first and foremost in the “separably tame setting” (introduced by Kuhlmann), which generalizes the work of Delon and others on separably algebraically maximal Kaplansky fields. Moreover, the existential portion of the theory of $\mathbb{F}_{p}((X))$ has been shown to be decidable, by myself and Fehm. On the hypothesis of resolution of singularities, Denef and Schoutens showed that the same is true in a stronger language, allowing the constant $X$; and with Dittmann and Fehm we have shown the same based on an a priori weaker hypothesis. More recent work, again with Fehm, has explored more closely the relationship between weakenings of local uniformization and axiomatizations of fragments of existential theories of power series fields.
I will attempt to survey this topic, including discussion of recent work on certain Hahn series fields.

We study bounds on the expected occupation density of diffusion processes whose drift and diffusion coefficients are not known precisely but are constrained to lie within prescribed intervals. Given a stopping horizon $\tau$, the goal is to determine the largest and smallest possible values of the density for the measure $$\Lambda(A)=\mathbb E\Big[\int_0^\tau \mathbf 1_{{X_t\in A}},dt\Big]$$ over all admissible diffusions, and to identify the extremal dynamics that attain these bounds. This problem admits a complete solution in one dimension, where occupation measures can be analysed through local-time techniques. In higher dimensions, the absence of such tools leads to a substantially richer and more challenging optimisation problem. I will present the one-dimensional theory, discuss what is known in several dimensions, and highlight a number of open questions. The analysis depends strongly on the choice of horizon: deterministic, exponentially distributed, and exit-time horizons give rise to related but distinct robust occupation problems. Beyond their intrinsic probabilistic interest, robust occupation bounds have applications in mathematical finance and stochastic control. They lead to model-independent bounds and hedging strategies for occupation-dependent derivatives and provide quantitative estimates on the worst-case performance loss of fixed control policies under model uncertainty.