Lecture 1 (Anton)
Poisson and symplectic structures, Hermitian matrices, coadjoint orbits of U(n).
Gelfand-Zeitlin integrable systems (regular fibers).
Exercise: collective integrable systems in the sense of Guillemin-Sternberg.
Lecture 2
Gelfand-Zeitlin integrable systems (singular fibers). (Jeremy)
Poisson-Lie groups. Poisson-Lie duality. (Anton)
Exercise: geometric quantization and Bohr-Sommerfeld fibers.
Lecture 3 (Anton)
Dual Poisson-Lie groups. Ginzburg-Weinstein Theorem (different proofs).
Tropicalization of dual Poisson-Lie groups. Berenstein-Kazhdan potential.
Lecture 4
Application: large Darboux charts on coadjoint orbits (Anton)
Toric degeneration and dense Darboux charts on coadjoint orbits (Jeremy)
In this talk, we explain a complex analysis approach to the lectures of Alekseev and Lane. In particular, we give an introduction to the Stokes phenomenon and the WKB approximation of meromorphic linear systems of ordinary differential equations, and prove that in a special case the Stokes phenomenon in the WKB approximation gives rise to Gelfand-Tsetlin systems. It is based on a joint work with Anton Alekseev and Yan Zhou.
The talk is based on a joint work with Maxim Kontsevich.
Moduli spaces of Stokes data is a generalization of the moduli spaces of local systems on topological surfaces. Precisely, the moduli space of vector bundles with meromorphic connection on a Riemann surface C has countably many components. These components are parametrised by the topological surface S underlying C, and a collection L of Legendrian links on S of a certain kind. The generalized Riemann-Hilbert correspondence identifies the component of the topological type (S, L) with a moduli spaces of Stokes data of the corresponding type.
I will explain that moduli spaces of framed Stokes data are shadows of non-commutative stacks of Stokes data, which carry a non-commutative cluster Poisson structure, equivariant under the wild mapping class group. The clusters are described by bipartite graphs on S. More generally, any bipartite graph on a surface S gives rise to a non-commutative cluster Poisson variety. In particular, if S is a torus, we get non-commutative cluster integrable systems.
There is a parallel story of non-commutative cluster A-varieties. They carry collections of cluster coordinates, which can be expressed via Gelfand-Retakh quasideterminants. In the rank 2 case we recover non-commutative surface cluster algebras of Berenstein-Retakh.
Quantum groups can be realized as quantized local systems, and as such admit quantum cluster structure. I will present this structure, and demonstrate its relation to the open quantum Toda and the Gelfand-Tsetlin integrable systems.