Cours fondamental 2 (Lie, GA, GC, GT)
Global Lie theory
Philip Boalch
Contact : boalch à imj-prg.fr
Pas de notes de cours prévues.
Langue du cours : anglais
Présentation
The exponential map for a complex Lie group can be viewed as the operation taking the monodromy of the rational connection
$$ A \frac{dz}{z} $$
on the punctured z-disk (where A is in the Lie algebra). This transcendental map has a vast collection of generalisations, the Riemann-Hilbert-Birkhoff maps, considering Lie algebra valued rational one-forms with any number of poles of any order. The Lie group then gets replaced by a global analogue of a Lie group, a wild representation variety. For example if we take the one-form to be of the form:
$$(A z+ B) \frac{dz}{z}$$
with A generic, then the wild representation variety is the Poisson variety underlying the Drinfeld-Jimbo quantum group.
The aim of this course is to study this "Lie theory over a curve", introducing the key players on both the additive and multiplicative sides. The spaces involved have fascinating geometry and are related to braid groups, infinite Weyl groups, quantum groups, Poisson geometry, hyperkahler geometry, Coulomb branches, Painlevé-type algebraic differential equations, the global Langlands program, generalizations of Teichmuller theory...
Contenu
- Algebraic connections on G-bundles on the punctured affine line and affine Kac-Moody algebras
- additive moduli spaces and Nakajima quiver varieties (quiver modularity theorem)
- wild Riemann surfaces, their fundamental groups and their admissible deformations. Relation to the Martinet-Ramis wild fundamental group
- multiplicative moduli spaces: representation varieties and character varieties (of wild Riemann surfaces)
- examples of: 1) global Weyl groups, 2) wild mapping class groups (such as quantum Weyl groups), 3) representation theory of wild character varieties
- quivers, diagrams and Cartan matrices in global Lie theory. Viewpoint of wild nonabelian motives.
Prérequis
Some elementary aspects of the courses: Théorie de Lie, représentations et géométrie I, Variétés algébriques, Géométrie différentielle et Riemannienne, Surfaces de Riemann will be useful.
The notes (\(\href{ https://webusers.imj-prg.fr/~philip.boalch/cours23/}{\text{here}}\)) might help too, as might some time spent playing with the following apps:
\(\href{https://webusers.imj-prg.fr/~philip.boalch/stokesdiagrams.html}{\text{Stokes Diagrams}}\)
and
\(\href{https://webusers.imj-prg.fr/~philip.boalch/fissiongraphs.html}{\text{Fission Graphs}}\)
(they will be explained in the course).
Bibliographie
- Roger Carter. Lie algebras of finite and affine type. CUP 2005
- M. Singer & M. Van der Put . Galois theory of linear differential equations. Springer 2003
- Pierre Deligne. Lettres à B. Malgrange 1978 et à J.-P. Ramis 1986. In: P. Deligne, B. Malgrange, and J.-P. Ramis, Singularités irrégulières, Documents Mathématiques, 5, Société Mathématique de France, Paris, 2007.
- P. Boalch. Symplectic Manifolds and Isomonodromic Deformations. Adv. in Math. 163 (2001) 137–205
https://webusers.imj-prg.fr/~philip.boalch/files/smid.pub.pdf
- P. Boalch. Geometry and braiding of Stokes data; Fission and wild character varieties. Annals of Math. 179 (2014) 301–365
https://arxiv.org/abs/1111.6228
- P. Boalch. Topology of the Stokes phenomenon. Proc. Symp. Pure Math. 103 (2021) 55–100
https://webusers.imj-prg.fr/~philip.boalch/files/tops.pdf