In many natural topological and geometric problems, one is led to study spaces that are very badly behaved from a classical point of view, such as the space of unitary representations of an infinite, non-abelian discrete group. Such spaces are quite often described well by noncommutative operator algebras: in particular, the Baum-Connes and coarse Baum-Connes conjectures predict that a good connection between the classical and noncommutative worlds is provided by higher index theory. These conjectures have many applications to geometry and topology, such as to the Novikov conjecture, and to the existence of positive scalar curvature metrics. It has recently become apparent, however, that coarse geometric properties associated to certain quotients of discrete groups and expanding graphs are obstructions to these conjectures: the investigator intends a systematic investigation of the geometric, analytic, and algebraic properties associated to these obstructions, and the phenomena thus allowed to exist. He also intends to apply realted ideas of coarse geometry and noncommuative geometry to prove classical index theorems associated to elliptic operators on symmetric spaces (with suitable boundary conditions).
Many natural spaces occurring in geometry have very bad properties from the point of view of classical mathematics. Inspired by quantum mechanics, one tries to describe these spaces using noncommutative mathematical structures - here 'noncommutative' means that the order in which one performs operations matters. If such a 'noncommutative description' is accurate in some precise sense, then a great deal of information about geometric spaces becomes available. It is known, however, that aspects of these descriptions fail for spaces called expanding graphs - for essentially the same reasons that make expanding graphs useful in computer science and the theory of networks. The investigator plans both to study, and to use, 'noncommutative descriptions' in borderline cases, particularly for spaces related to expanding graphs and the discrete groups that can be used to construct them. A better understanding of exotic properties of discrete groups, and thus of their connections to other areas of mathematics and computer science, is also a central goal.
