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).

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