Related papers: The Gromov-Lawson index and the Baum-Connes assemb…

The surgery technique of Gromov and Lawson may be used to construct families of positive scalar curvature metrics which are parameterised by Morse functions. This has played an important role in the study of the space of metrics of positive…

We study reductions well suited to compare structures and classes of structures with respect to properties based on enumeration reducibility. We introduce the notion of a positive enumerable functor and study the relationship with…

We study the properties of certain graphs involving the sums of primes. Their structure largely turns out to relate to the distribution of prime gaps and can be roughly seen in Cram\'er's model as well. We also discuss generalizations to…

We show that results about spaces or moduli spaces of positive scalar curvature metrics proved using index theory can typically be extended to non-negative scalar curvature metrics. We illustrate this by providing explicit generalizations…

We study Gauss curvature for random Riemannian metrics on a compact surface, lying in a fixed conformal class; our questions are motivated by comparison geometry. Next, analogous questions are considered for the scalar curvature in…

We document some versions, in real K-theory, of well-known properties of the coarse assembly map in complex K-theory. These results are well-known, but difficult to find in the literature.

We introduce Riemannian metrics of positive scalar curvature on manifolds with Baas-Sullivan singularities, prove a corresponding homology invariance principle and discuss admissible products. Using this theory we construct positive scalar…

Inspired by an analytic construction of Chang, Weinberger and Yu, we define an assembly map in relative geometric $K$-homology. The properties of the geometric assembly map are studied using a variety of index theoretic tools (e.g., the…

We present an alternative approach to the result of Guentner, Higson, and Weinberger concerning the Baum-Connes conjecture for finitely generated subgroups of SL(2,C). Using finite-dimensional methods, we show that the Baum-Connes assembly…

In this paper, we prove a quantitative relative index theorem. It provides a conceptual framework for studying some conjectures and open questions of Gromov on positive scalar curvature. More precisely, we prove a $\lambda$-Lipschitz…

We prove that the coarse assembly maps for proper metric spaces which are non-positively curved in the sense of Busemann are isomorphisms, where we do not assume that the spaces are with bounded coarse geometry. Also it is shown that we can…

We explain how to derive largeness constraints in scalar curvature geometry using some basic splitting results and the potential theory on singular area minimizing hypersurfaces. This includes a variety of results like the non-existence of…

For a proper action by a locally compact group $G$ on a manifold $M$ with a $G$-equivariant Spin-structure, we obtain obstructions to the existence of complete $G$-invariant Riemannian metrics with uniformly positive scalar curvature. We…

In the first part we use Gromov's K--area to define the K--area homology which stabilizes into singular homology on the category of pairs of compact smooth manifolds. The second part treats the questions of certain curvature gaps. For…

Let $\Gamma$ be a discrete group. Assuming rational injectivity of the Baum-Connes assembly map, we provide new lower bounds on the rank of the positive scalar curvature bordism group and the relative group in Stolz' positive scalar…

We introduce the notion of one-sided mapping cones of positive linear maps between matrix algebras. These are convex cones of maps that are invariant under compositions by completely positive maps from either the left or right side. The…

We study in this paper the maximal version of the coarse Baum-Connes assembly map for families of expanding graphs arising from residually finite groups. Unlike for the usual Roe algebra, we show that this assembly map is closely related to…

We describe a geometric counterpart of the Baum-Connes map for the p-adic group GL(n).

We prove the Gromov-Lawson-Rosenberg conjecture for cocompact Fuchsian groups, thereby giving necessary and sufficient conditions for a closed spin manifold of dimension greater than four with fundamental group cocompact Fuchsian to admit a…

We study the statistical properties of piecewise expanding maps in the general setting of metric measure spaces. We provide sufficient conditions for exponential mixing of such systems with explicit estimates on the constants. We also…