SonstigeVeranstaltungenMI An overview on Lie pseudogroups and geometric structures |  |
Speaker:Francesco Cattafi, Utrecht University
Organizer:Mathematisches Institut
Details:
A Lie pseudogroup $\Gamma$ is a collection of locally defined diffeomorphisms arising as solutions of a PDE. This object gives rise to the notion of $\Gamma$-structure on a manifold $M$: it is a maximal atlas whose changes of coordinates take values in $\Gamma$.
For instance, the set of local symplectomorphisms of the canonical symplectic structure of $\mathbb{R}^{2n}$ forms a Lie pseudogroup, and the associated $\Gamma$-structure is a symplectic structure on $M$. More generally, starting from a Lie subgroup $G \subseteq GL(n,\mathbb{R})$, one can define a Lie pseudogroup $\Gamma_G$, and see that a $\Gamma_G$-structure coincides with the standard notion of an integrable $G$-structure on $M$.
Nevertheless, a number of geometric structures can be described by Lie pseudogroups and not by Lie groups. This leads us to a natural question: what is the counterpart of a non-integrable $G$-structure in the Lie pseudogroup world (which we call "almost $\Gamma$-structure")? And when does an almost $\Gamma$-structure come from a $\Gamma$-structure?
In this talk we are going to review these notions and provide an answer to the questions sketched above. In particular, we present a new characterisation of formal integrability in the setting of $\Gamma$-structures; this will be obtained by introducing the concept of principal Pfaffian bundle and studying its prolongations to higher orders. For doing that, we draw inspiration from similar results for PDEs on jet bundles and for $G$-structures, which we are going to recover. This is joint work with Marius Crainic.
For instance, the set of local symplectomorphisms of the canonical symplectic structure of $\mathbb{R}^{2n}$ forms a Lie pseudogroup, and the associated $\Gamma$-structure is a symplectic structure on $M$. More generally, starting from a Lie subgroup $G \subseteq GL(n,\mathbb{R})$, one can define a Lie pseudogroup $\Gamma_G$, and see that a $\Gamma_G$-structure coincides with the standard notion of an integrable $G$-structure on $M$.
Nevertheless, a number of geometric structures can be described by Lie pseudogroups and not by Lie groups. This leads us to a natural question: what is the counterpart of a non-integrable $G$-structure in the Lie pseudogroup world (which we call "almost $\Gamma$-structure")? And when does an almost $\Gamma$-structure come from a $\Gamma$-structure?
In this talk we are going to review these notions and provide an answer to the questions sketched above. In particular, we present a new characterisation of formal integrability in the setting of $\Gamma$-structures; this will be obtained by introducing the concept of principal Pfaffian bundle and studying its prolongations to higher orders. For doing that, we draw inspiration from similar results for PDEs on jet bundles and for $G$-structures, which we are going to recover. This is joint work with Marius Crainic.
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Type:Talk
Language:English
Category:Research
Host:JProf. Madeleine Jotz Lean
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Direct link to event:https://events.goettingen-campus.de/event?eventId=17597
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