Automorphisms of the Boutet de Monvel algebra

Automorphisms of the Boutet de Monvel algebra

Speaker：

Ryszard Nest (Copenhagen University)

Time：

Thur., 14:00-15:00, Nov. 28, 2024

Venue：

C548, Shuangqing Complex Building A

Online：

Zoom Meeting ID: 271 534 5558

Passcode: YMSC

Abstract:

In a remarkable work, Duistermaat and Singer in 1976 studied the algebras of all classical pseudodifferential operators on smooth (boundaryless) manifolds. They gave a description of order preserving algebra isomorphism between the algebras of classical pseudodifferential operators of two manifolds. The subject of this talk is the generalisation of their results to manifolds with boundary. The role of the algebra of pseudodifferential operators that we are interested in is the Boutet de Monvel algebra.

The main fact of life about manifold with boundary is that vector fields do not define global flows and the "boundary conditions" are a way of dealing with this problem. The Boutet de Monvel algebra corresponds to the choice of local boundary conditions and is, effectively, a non-commutative completion of the manifold. One can think of it as a parametrised version of the classical Toeplitz algebra as a completion of the half-space.

What appears in the study of automorphisms are Fourier integral operators and we will try to explain their appearance - both in boundaryless and boundary case. as it turns out, the non-trivial boundary case introduces both some complications but also some simplifications of the analysis involved, Once this is done, the analysis that we need reduces to a high degree to relatively classical results about automorphisms and homology of the Toeplitz algebra and some basic facts from K-theory.

This is a joint work in progress with Elmar Schrohe.