06/09/2016, 10:20 — 11:00 — Amphitheatre Pa2, Mathematics Building
Rui Loja Fernandes, University of Illinois at Urbana-Champaign

A classification of non-commutative integrable systems

A non-commutative integrable system (NCIS) on a symplectic manifold $(X^{2n}, \Omega)$ is given by a collection of functions $\{f_1, \dots, f_k\}$ where $(k\geq n)$, satisfying the following two assumptions:

1. Involutivity: the first $r=2n-k$ functions Poisson commute with all $k$ functions:

   \[\{f_i,f_j\}=0,\qquad (i=1,\dots,r; j=1,\dots,k).\]
2. Independence: the functions are independent almost everywhere: \[\operatorname{d} f_1\wedge \dots \wedge\operatorname{d}f_k\neq 0 \qquad \text{on a dense open set}.\]

When $k=n$ one recovers the classical notion of a commutative integrable system (CIS). The same way a CIS is related to a Lagrangian fibration, NCIS are related to isotropic fibrations. In this lecture I will explore this relationship and some beautiful connections with Poisson geometry, integral affine geometry and symplectic groupoids, leasing to a classification of regular NCIS.

This talk is based on various ongoing collaborations with Marius Crainic (Utrecht), David Martinez Torres (PUC-Rio), Daniele Sepe (UFF-Rio), Camille Laurent-Gengoux (Metz) and Pol Vanhaeck (Poitiers).