# Order-preserving isomorphisms of algebras of pseudo-differential operators

#### 07.07.2022, 16:00  –  2.09.0.14 Forschungsseminar Differentialgeometrie

Alessandro Contini (Leibniz University Hanover)

In 1971 Egorov proved his famous Theorem, stating that conjugating a pseudo-differential operator with an invertible Fourier Integral Operator produces a new $$\Psi$$DO with the same principal symbol. This implies in particular that the map $$\varphi(A)\colon=FAF'$$ for $$F'$$ a parametrix of the elliptic FIO $$F$$ is an order-preserving automorphism of the algebra $$\Psi(M)$$ for $$M$$ a manifold. In the subsequent years the following question (a kind of converse to Egorov's result) was addressed: can we characterize all the order-preserving algebra isomorphisms between $$\Psi(X)$$ and $$\Psi(Y)$$ for any manifolds $$X,Y$$? Positive answers were given first by Duistermaat and Singer (with refinements by Mathai and Melrose) in the Hörmander calculus, then by Christianson for semi-classical operators, and finally by Battisti-Coriasco-Schrohe for the Boutet de Monvel algebra. In this talk I shall recall and give an overview of the above mentioned concepts and results. Then I shall dedicate some time to present the so-called SG-calculus of pseudo-differential operators, and finally I will outline some ideas and problems encountered up until now. This is based on work-in-progress during my current time as a PhD student with Elmar Schrohe.

zu den Veranstaltungen