The talk will describe some recent geometric constructions related to the algebraic spectrum of the integers and the adele class space of the rationals which are part of a joint work with A. Connes.

In this talk, I will discuss our resent work on a class of essentially normal operators by using the geometry method from the Cowen-Douglas theory and a Brown-Douglas-Fillmore theorem in the Cowen-Douglas theory. More precisely, the Chern polynomials and the second fundamental forms are the similarity invariants (in the sense of Herrero) of this class of essentially normal operators.

We introduce a notion of 'local decomposability' for a C*-algebra, inspired by the theory of nuclear dimension (due to Winter and Zacharias) and of dynamical complexity (due to Guentner, Yu and the speaker). We derive the existence of a sort-of controlled Mayer-Vietoris sequence in this setting, inspired by work of Oyono-Oyono and Yu in controlled K-theory (although not using that language), and give applications to Baum-Connes theory and to the Künneth formula.

The Novikov conjecture is an important problem in higher dimensional topology. It claims that the higher signatures of a compact smooth manifold are invariant under orientation preserving homotopy equivalences. The Novikov conjecture is a consequence of the strong Novikov conjecture in the computation of the K-theory of group C*-algebras. In this talk, I will talk about the Novikov conjecture for groups which are extensions of coarsely embeddable groups.

This talk is on aspect of my general project with Carl Jockusch on “the coarsification of computability theory”, that is, bringing the asymptotic-generic point of view of geometric group theory into the theory of computability. Classically, computability theory studies Turing degrees, that is, equivalence classes of subsets of N which are computationally equivalent. Coarse computability studies how closely arbitrary subsets of N can be approximated by computable sets. The idea of coarse computabilty leads to a natural definition of the closure of a Turing degree in the space S of coarse similarity classes of subsets of N with the Besicovich metric. It turns out that S is an interesting space. We will discuss interactions of the topology of S and properties of Turing degrees.

Roe defined a localised version of the coarse index of an elliptic operator that is invertible outside a subset Z of the manifold M it is defined on. An equivariant version of this index was defined for proper and free actions by discrete groups by Xie and Yu. With Guo and Mathai, we extended this to proper actions by any locally compact group G. If Z/G is compact, then this index takes values in the K-theory of the group C* algebra of G, and generalises the Baum-Connes analytic assembly map. It also generalises an equivariant index of Callias-type operators constructed earlier by Guo. Another special case is an equivariant index for proper, cocompact actions on manifolds with boundary, generalising the Atiyah-Patodi-Singer (APS) index and its equivariant version. With Bai-Ling Wang and Hang Wang, we obtained an equivariant APS index theorem in this context. Using a version for maximal group C*-algebras and Roe algebras, we obtain a link with an index on invariant sections defined earlier with Mathai.

I will report on joint work in progress with N. Christopher Phillips. In 2010, Giol and Kerr published a construction of a minimal dynamical system whose associated crossed product has positive radius of comparison. Subsequently, Phillips and Toms conjectured that the radius of comparison of a crossed product should be roughly half the mean dimension of the underlying system. Upper bounds were obtained by Phillips, Hines-Phillips-Toms and very recently by Niu, however there were no results concerning lower bounds aside for the examples of Giol and Kerr. In the non-dynamical context, work of Elliott and Niu suggests that the right notion of dimension to consider is cohomological dimension, rather than covering dimension (notions which coincide for CW complexes). Motivated by this insight, we introduce an invariant which we call "mean cohomological independence dimension" (more precisely, a sequence of such invariants), for actions of countable amenable groups on compact metric spaces, which are related to mean dimension, and obtain lower bounds for the radius of comparison for crossed products in terms of this invariant.