# Junior Seminar

**Venue:** HIM lecture hall, Poppelsdorfer Allee 45

Organizers: Simon Brain, Jens Kaad, Bram Mesland

## Friday, September 5

16:30 - 18:00 Welcome Meeting

Abstract: Welcome meeting of the junior participants (students and postdocs).

## Monday, October 20

10:30 - 11:30 Francesca Arici: Generalized Crossed Products and Takai Duality

Abstract: There has been a lot of interest lately in generalized crossed product algebras. This notion is deeply related to that of Cuntz-Pimsner algebras and generalizes both crossed products by the integers and Cuntz-Krieger algebras. In this talk I will give the definition of the crossed product algebra by a C* correspondence and exhibit examples in the particular case when the C* correspondence is a self Morita equivalence. Finally, I will show that it is possible to prove an analogue of Takai duality in this more general setting. This is mainly based on work by B. Abadie and collaborators.

## Friday, October 24

16:30 - 17:30 Iain Forsyth: Spectral triples, Symmetric Spectral Triples and Relative Spectral Triples

Abstract: Spectral triples are a geometric way to obtain Fredholm modules and hence K-homology classes. There have been some attempts in the literature to slightly weaken the definition of a spectral triple, but recent work by Bram Mesland, Adam Rennie and myself shows that this seemingly harmless modification results in a failure to obtain a Fredholm module. Such a problem in particular occurs when one attempts to construct a spectral triple for the algebra of smooth functions on a manifold with boundary. I will discuss how one can instead construct spectral triples for the algebra of functions vanishing on the boundary, or the algebra of functions constant on the boundary. I will also discuss symmetric spectral triples (due to Hilsum) and relative spectral triples, which are generalisations of spectral triples particularly suited to manifolds with boundary.

## Friday, October 31

16:30 - 17:30 Christopher Bourne: The Bulk-Edge Correspondence for the Quantum Hall Effect in Kasparov Theory

Abstract: The bulk-edge correspondence involves equating topological properties of the interior of a physical system with topological properties of its edge or boundary. In this talk I will show how the work of Kellendonk, Schulz-Baldes and others on the bulk-edge correspondence for the quantum Hall effect can be recast into the language of Kasparov modules and KK theory. This makes the general method we use more flexible and potentially applicable to other physical systems. This is joint work with Alan Carey and Adam Rennie.

## Friday, November 21

16:30 - 17:30 Niek de Kleijn: Fedosov construction of formal deformation quantization

Abstract: We will give a brief introduction to and motivation of the general set up of deformation quantization. In particular we will discuss the ideas behind the Fedosov construction of deformation quantization of symplectic manifolds. Of course the rigorous construction will also be presented. As an application we will present the construction of induced symplectomorphisms and symplectic actions of (discrete) groups on deformation quantization. Finally (if time permits) we would like to give some overview of the classification of such group actions.