Combinatorics Seminar 2017

Speaker: Nguyen Tien Zung, Universit́e Paul Sabatier

Date: Thu, Jun 08, 2017

Time: 15:00 - 16:00

Venue: Middle Meeting Room

Title: Convexity in examples, from Lie theory, symplectic geometry and integrable Hamiltonian systems


Convexity is an interesting and important phenomenon in Lie theory and symplectic geometry, because it links geometric properties to combinatorial invariants and, at the same time, it provides a framework for the classification of dynamics with symmetry. This talk is based on joint work with Tudor Ratiu and consists of three parts.

The first part is a brief review of classical convexity theorems: the Schur-Horn theorem about the diagonals of Hermitian matrices, Kostant's convexity theorem in Lie theory, Atiyah-Guillemin-Sternberg's and Kirwan's theorems on convexity of Hamiltonian compact Lie group actions on symplectic manifolds, and various extensions.

The second part reviews separate convexity work of Ratiu and myself. Ratiu worked on several aspects, including convexity of Poisson-Lie group actions and convexity of the infinite-dimensional group of area-preserving diffeomorphisms of the annulus. I obtained linearization and convexity results for quasi-symplectic groupoids and their actions, which cover other convexity results as special cases, including convexity for group-valued momentum maps.

The third part is a report on our joint recent work on convexity of integrable Hamiltonian systems on (pre)symplectic manifolds, also in the presence of focus-focus singularities, which often appear in physical systems. A surprising phenomenon happens here: the locally-flat integral affine spaces can be bent by the monodromy created by these singularities into spaces which are locally convex but globally non-convex, and which can even contain "affine black holes".

Slides: View slides