Computing homological invariants of dynamical systems and quantum groups

The pervasive role of algebraic topology in mathematics is proof of
the powerful effects that homological invariants produce in the
development of the discipline. Extending these techniques beyond
the category of topological spaces, in order to include "quantized"
systems arising from dynamical systems and (quantum) groups, is
going to be extremely useful to make fast progress in these fields.
The framework of operator algebras and noncommutative geometry
is extremely well-suited for these developments and has already
been applied with some success. The goal of this proposal is to
further develop these homological techniques by supporting them
with novel methods based on triangulated categories, homotopy
theory, and index theory. The research problems tackled in this
proposal are deeply related to important topics which attracted a
great deal of interest in the mathematical community. For example,
we study the celebrated Baum-Connes conjecture (for both
groupoids and quantum groups) through a relatively unexplored
perspective and relate it to the computation of K-theoretic and
homological invariants for notable dynamical systems (e.g., Smale's
Axiom A diffeomorphisms). This research will provide mathematicians
with both conceptually new approaches and powerful computational
tools. Some of these results are relevant not only for pure
mathematics, but also for solid-state physics and quantum
information theory..