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**Title. **Pushing index theorems into the Sobolev

**Abstract. **The classification table of topological condensed matter systems is often and erroneously said to reflect the K-theories of the underlying algebras of observables. In fact, the groups appearing in that table are much smaller than the K-groups. In this talk, I will explain how the classification table is rather related to index theorems derived from the pairing of the relevant K-theories with Alain Connes cyclic cohomology. As an example, I will work out the case of class A of classification table in arbitrary even dimension, following [1]. The algebra of observables is the disordered non-commutative torus and the task is to prove an index theorem for the top pairing which can be pushed into a certain non-commutative Sobolev space, which turned out to cover the regime of strong disorder, relevant to the experiments. As we shall see, the algebraic side of the program supply guidance but it is analysis that eventually finishes the job.

1. E. Prodan, H. Schulz-Baldes, Bulk and Boundary Topological Invariants:

From K-Theory to Physics, (Springer, Berlin, 2016).