Program
Title: Poisson-Lie groups and pseudo-differential symbols.
Abstract: In this talk I will introduce the notion of finite-dimensional Poisson-Lie group focusing in giving two different but equivalent infinitesimal descriptions of them. Namely, we show how to obtain the Lie bi-algebra structure as well as the Manin triple associated to a Poisson-Lie group. Finally, I will define the group of pseudo-differential symbols and show that it admits a natural Poisson-Lie group structure.
Files: Lecture notes - preliminary version. These notes will be merged with those of João Felipe Pereira and Kelvyn Emmanoel.
Title: Noether's theorems and the relation of conserved quantities and symmetries on classical mechanics systems.
Abstract: In this talk the Noether theorems will be presented and discussed on both Lagrangian and Hamiltonian frameworks. This result plays a major role in the description of classical mechanics systems by relating conserved quantities with the symmetries groups that act on the system. In particular, it was present in the cases studied in the latter part of this course. Naturally, the descriptions will be accomplished through a geometric approach.
Files: Lecture notes.
Observation: This seminar was presented as a lecture for the course Lie groups and fluid dynamics, on 05/21/2022.
Title: Stability of stationary points on Lie algebras.
Abstract: In this talk, I will present a stability criterion established by Vladimir Arnold for the Euler equation on an arbitrary Lie Algebra.
Title: Discrete dynamos in two dimensions: cat map, Smale's horseshoe and homoclinic points.
Abstract: In this seminar we will learn concepts motivated by magnetodynamics: what is a dynamo, origin of the dynamo equation, example of the discrete dynamo in dimension two (the cat map), in short we will learn the relation of homoclinic points with the existence of Smale's horseshoe.
Title: (Not) wandering solutions of Euler equations
Abstract: Poincaré's recurrence theorem claims that every trajectory of the Euler equation on a finite-dimensional Lie algebra returns repeatedly to the vicinity of its initial position. However, the Euler equation for an ideal fluid has an infinite-dimensional associated Lie algebra and it is possible to construct a counter-example to show it does not enjoy the recurrence property.
Title: Slice theorem for the space of Riemannian metrics.
Abstract: In this talk will be discussed a version of the slice theorem for the space of Riemannian metrics equipped with the action of the group of diffeomorphisms. This will be done by following the point of view provided in "Short survey on the existence of slices for the space of Riemannian metrics" which proposes an approach more geometric than the original one by Ebin around 1970.
Files: arXiv preprint.
Observation: This seminar will be presented as a lecture for the course Lie groups and fluid dynamics, on 05/28/2022.
Title: Central Extensions of Double Loop Groups and Their Lie Algebras
Abstract: We will present the theory of central extensions of elliptic curves in Lie Groups and Lie Algebras and the so-called elliptic Algebras. The main theorem relates the coadjoint orbits of double loop groups and holomorphic G-bundles over the curve.
Title: Integrability, symplectic reduction and Calogero-Moser systems.
Abstract: Calogero-Moser systems are a family of Hamiltonian systems based on Weierstrass elliptic functions and their degenerancies. In this talk we shall see how those systems can be represented as symplectic reductions on some classes of Lie algebras, helping us to show that they are integrable and relating those (a priori unrelated) classes of Lie algebras.
Title: Pseudodifferential symbols and their central extensions
Abstract: In this seminar, we will introduce the notion of differential symbols on an associative algebra and use it to motivate the extended algebra of pseudodifferential symbols. We then study its most important properties and centrally extend the new algebra via a suitable Lie 2-cocycle. Examples are given in the case where the ambient associative algebra is the group of diffeomorphisms on the circle.
Files: Lecture notes - preliminary version. These notes will be merged with those of Fabricio Valencia and Kelvyn Emmanoel.
Title: Plasma physics.
Abstract: The plasma state can be described by the Maxwell-Vlasov equations. The goal of this talk is to introduce such a state as an infinite-dimensional Hamiltonian system over a phase space which is determined by the reduction of a Poisson manifold.
Title: Moduli Spaces of Flat Connections over Surfaces.
Abstract: Given a compact and orientable surface X and a Lie group G, the space of connections on the trivial principal G-bundle over X admits a symplectic structure. The group of gauge transformations of the bundle acts on this space preserving that structure. In this talk we will see some properties of these actions when X has a boundary or not. As a result, we obtain the moduli space of flat connections modulo gauge transformations, which will be a Poisson manifold with singularities. We will present ways to obtain these spaces, some examples and finally, their symplectic foliations.
Title: Pseudodifferential symbols and Manin triples.
Abstract: Continuing the study of the Lie algebra of pseudodifferential symbols, we define a Manin triple and prove that this algebra, as well as another extended version of it via a semidirect product, form Manin triples. For the next step, we show that, in the finite-dimensional context, Manin triples are in one-to-one correspondence with another object: Lie bialgebras, which are the infinitesimal version of a Poisson-Lie group, paving the way to construct the Poisson-Lie group of pseudodifferential symbols.
Files: Lecture notes - preliminary version. These notes will be merged with those of João Felipe Pereira and Fabricio Valencia.
Title: Central extensions of the gauge algebra.
Abstract: In this talk it will be introduced the construction of a cocycle, which in turn will give rise a central extension, for the gauge algebra of a principal bundle. It will be done first for a trivial bundle over a compact manifold where the gauge algebra is isomorphic to the Lie algebra of smooth maps from the manifold to the Lie algebra of its structural group. Finally, the argument will be extended for the gauge algebra of an arbitrary principal bundle provided the construction of a covariant cocycle.