Virtual Workshop C*-algebras, K-theories and Noncommutative Geometries of Correlated Condensed Matter Systems: May 17-21, 2021

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Organized by: Emil Prodan, Anton Kapustin and Nigel Higson

The uncorrelated topological phases of matter are relatively well understood by now, even in the regime of strong disorder. Since C*-algebras, K-theory and noncommutative geometry (NCG) have been critical to that understanding, they might also expected to supply a mathematical framework for the study of phases of gapped many-body systems, and in particular provide a general and precise formulation and characterization of the bulk-boundary principle, as well as a rigorous definition and classification of Short-Range Entangled (SRE) phases of both fermions and bosons. However, while the K-theory/NCG formalism has been able to deal with aperiodic aspects, it has encountered fundamental difficulties when faced with interactions.

For example, the C*-algebra of observables for fermionic interacting systems, which is the universal algebra generated by the canonical anti-commutation relations (CAR) over a separable Hilbert space, is extremely simple from the K-theoretic point of view. But the Hamiltonian does not belong to this C*-algebra; rather it implements a one-parameter group of outer automorphisms. So, mathematically, we are presented with a triple consisting of the algebra, the time evolution and a time-invariant state. Given data of this kind, the task is to formulate a correct classification program based on K-theory and NCG, which explains the experimental observations (for example the existence of Hall-plateaus at fractional values), has the power to predict new ones, and supplies a natural framework to compute numerical invariants associated with physical observations.

The aims of this workshop will be as follows:

1. Demystify the K-theory/NCG formalism for the physics community. The workshop will include pedagogical lectures covering basic concepts in C*-algebras and standard examples, explicit computations in condensed matter systems, and applications that have led to a deep understanding of the dynamical, spectral and topological properties of condensed matter systems.

2. Present an overview of the major programs in K-theory and NCG. Pedagogical as well in-depth lectures will be given covering the main programs in K-theory and NCG, including efforts towards a constructive Kasparov K-theory, the Baum-Connes conjecture, and index theory for non-commutative spaces.

3. Examine correlated models of condensed matter through the prism of C*-algebras and K-theory. The workshop will include overviews of representative physical models and various programs of classification, as well as examples of topological invariants of many-body systems, progress on K-theory of quantum groups, the functorial aspects of KK-theory, and the state of the art on CAR algebras.