This talk offers a concise overview of Lie systems, commemorating the 16th anniversary of my PhD defense. I will trace the historical development of Lie systems, beginning with foundational work by Guldberg, Vessiot, and Lie, which had a relevant moment in 1893 with the celebrated Lie theorem. This will be followed by the analysis of the contributions from Winternitz and the CRM school in Canada. My presentation will then explore the geometric approach (Carinena, Marmo, Grabowski) developed at the begining of my PhD. Then, I will explore the developments during my PhD thesis culminating with the coalgebra method (developed with my coworkers), deformation theory, Lie systems related to geometric structures, and other developments that followed after that. I will finish the newer approaches to Lie systems and their generalisations: stochastic and super Lie systems, discrete Lie systems, and Lie groupoid methods in Lie systems. I will also pay special attention to the results by Marmo, Carinena, Grundland, Ballesteros, Herranz, Fernandez-Sainz, Sardon, Campoamor-Stursberg, Carballal, Odzijewicz, Ibragimov, etc.

join us at 15:15

Robet Wolak (Jagiellonian University)

Sasakian geometry and foliations (provisional title)

join us at 15:15

Tomasz Sobczak (KMMF)

Introduction to PDE Lie systems

This talk aims to present the theory of systems of first-order partial differential equations that admit a (nonlinear) superposition rule: the so-called PDE Lie systems. These systems are characterized by the property that the partial derivatives of their solutions depend only on the independent and dependent variables, and their general solution can be expressed as a function of a family of particular solutions together with several constants. In particular, we will discuss the Lie–Scheffers theorem and the Lie group approach to PDE Lie systems. We will present the main features of these systems as well as a range of classical and novel results concerning Bäcklund transformations, Lax pairs, Floquet theory, geometric phases, conditional symmetries, and further potential applications in hydrodynamical systems.