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# Martin Grensing

INdAM-COFUND Incoming fellow between 2012-02-01 and 2012-10-31

### Project information

#### Title

Bivariant K-theory, Index Theory and Mathematical Physics

#### Partner organisation

Center for Mathematics and Theoretical Physics

Via della Ricerca Scientifica, 1

00133 Roma, Italy

#### Abstract

The project is centered around bivariant K-theory and index theory. We will study in how far it is possible to define such theories using methods from stable homotopy theory. It is, in particular, to be expected that this gives a way to generalise

Kasparov’s KK to more general categories, such as locally convex algebras. We will further investigate index theory in terms of bivariant K-theory, based on the notion of locally convex Kasparov modules we have introduced earlier. This will eventually lead to a theory based on infinite-dimensional geometries.

This will also allow to interpret index theory in terms of bivariant K-theory, directly on the algebras of smooth functions. In the classical theory, all of this has been developed by Kasparov, but the Chern character is of little use because of the fact that cyclic periodic cohomology is in a sense degenerate. Thus in order to prove index theorems in a “universal” manner, one has to develop either a version of periodic cyclic cohomology for C*-algebras, or bivariant K-theories adopted to a larger class of algebras. The first approach has been followed by M. Puschnigg, a first attempt to attack the second is Cuntz’ kk-theory. However, it seems this theory might not be universal among split exact, diffeotopy invariant and stable functors.

### Fellow information

#### Previous positions, awards

No previous positions recorded.

#### Publications, preprints, other works

- M. Grensing,
*KK, stable homotopy theory and unstable Ext-theory*, submitted - O. Gabriel, M. Grensing,
*Six-Term Exact Sequences for Smooth Generalized Crossed Products*, preprint (2011); arXiv:1111.2154, to appear in the Journal of Noncommutative Geometry - M. Grensing,
*Universal cycles and homological invariants of locally convex algebras*, preprint (2011); arXiv:1103.6243, to appear in the Journal of Functional Analysis - M. Grensing,
*The Thom isomorphism in bivariant K-theory*, preprint (2011); arXiv:1103.6245, submitted - O. Gabriel, M. Grensing,
*Exact sequences for locally convex subalgebras of Pimsner algebras with an application to Quantum Heisenberg Manifolds*, preprint (2010); arXiv:1011.6238 - M. Grensing,
*A note on Kasparov products*, preprint (2011); arXiv:1103.6244, to appear in the Journal of K-theory - M. Grensing,
*The Exact Embedding Theorem*, Diploma thesis; available at http://www.martin.grensing.net/articles/M.%20Grensing-The%20Exact%20Embedding%20Theorem.pdf

#### Conferences, schools, other events

*10th GEOMETRY AND PHYSICS CONFERENCE “Quantum Geometry”*

6-10 August 2012, Anogia, Greece*Noncommutative geometry, index theory and applications*

11-15 June 2012, Cortona, Italy