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Winter Lectures 2025 by Alessandra Frabetti
Winter Lectures 2025 by Alessandra Frabetti
On the 18th of December, 2025, Alessandra Frabetti (Université Claude Bernard Lyon 1) gave two special lectures.
Talk 1 — 11:00–12:30
Transport maps on manifolds using groupoids and direct connections
In the theory of Regularity Structures, local solutions of (singular and stochastic) PDEs at different points of a base manifold are related by means of some transport maps. On non-flat manifolds, such operators are best described by Lie groupoids, a bi-fibred version of Lie groups where each element has a source and a target point on the manifold. After a short review of two
motivating examples, we shall introduce the basic mathematical tools for such a geometric approach: groupoid actions on vector bundles and direct connections on groupoids giving rise to transport maps.
Talk 2 — 2:00–2:45 pm & 3:00–3:45 pm
(with a short break for coffee, cookies, and conversation)
Direct connections on jet groupoids and regularity structures
Solutions of PDEs obtained by means of some local Taylor expansion are geometrically naturally described in terms of jet bundles. In order to relate such local solutions, we consider direct connections on jet groupoids. In this talk we shall present a natural jet prolongation of direct connections to jet groupoids. We shall also provide an example of a direct connection on a jet groupoid which does not arise in the context of Regularity Structures.
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