Miniworkshop on noncommutative geometry and pseudodifferential operators on the occasion of Magnus Fries public PhD-defense. 

13.15-14.00 Matthias Lesch (University of Bonn)

Title: Rearrangement Lemma, divided differences and the multivariate holomorphic functional calculus

The so called Rearrangement Lemma is a technical device in the context of heat trace expansions on noncommutative spaces resp. for operators with noncommutative leading symbol.

A couple of years ago I gave a systematic treatment and discussed the link to the classical divided difference formalism.

In this talk I would like to recast the issue from the point of view of the multivariate holomorphic functional calculus.

The latter has a rich history and is of some interest in its own as it is by no means just a straightforward generalization of the usual one-variable case. I will review this history and give some elementary applications to noncommutative versions of Newton interpolation and Taylor formulas as well as a holomorphic version of the rearrangement lemma.

The talk is based on a recent paper with Luiz Hartmann from Sao Carlos, Brazil to be published in Doc. Math.

14.15-15.00 Jens Kaad (University of Southern Denmark)

Title: Spectral localizers in KK-theory.

In this talk we compute the index homomorphism of even K-groups arising from a class in even KK-theory via the Kasparov product. Due to the seminal work of Baaj and Julg, under mild conditions on the C*-algebras in question, every class in KK-theory can be represented by an unbounded Kasparov module. We then describe the corresponding index homomorphism of even K-groups in terms of spectral localizers. This means that our explicit formula for the index homomorphism does not depend on the full spectrum of the abstract Dirac operator D, but rather on the intersection between this spectrum and a compact interval. The size of this compact interval does however reflect the interplay between the K-theoretic input and the abstract Dirac operator. Since the spectral projections for D are not available in the general context of Hilbert C*-modules we instead rely on certain continuous compactly supported functions applied to D to construct the spectral localizer. In the special case where even KK-theory coincides with even K-homology, our work recovers the pioneering work of Loring and Schulz-Baldes on the index pairing.

15.00-15.45 Coffee break

15.45-16.30 Ada Masters (Lund University)

Title: Conformal symmetry of the Podleś sphere

I will describe the notion of conformal group equivariance for unbounded KK-theory and its generalisation to quantum groups, extending the work of Baaj and Skandalis in the bounded picture. This requires the new notion of matched operator on a Hilbert C*-module, the collection of which forms a pro-C*-algebra. I will focus on the example of the Podleś sphere, whose equivariance at the level of bounded KK-theory was demonstrated by Nest and Voigt.

16.45-17.30 Joachim Toft (Linnæus University)

Title: Fractional Fourier transform, harmonic oscillator propagators and Strichartz estimates

Using the Bargmann transform, we give a proof of that harmonic oscillator propagators and Fractional Fourier Transforms (FFT) are essentially the same. We deduce continuity properties for such operators on modulation spaces, and apply the results to prove Strichartz estimates for the harmonic oscillator propagator when acting on modulation spaces. We also show that general forms of fractional harmonic oscillator propagators are continuous on suitable on so-called Pilipovic spaces and their distribution spaces. Especially we show that FFT of any complex order can be defined, and that these transforms are continuous on strict Pilipovic function and distribution spaces. The talk is based on a joint work with Divyang Bhimani and Ramesh Manna.