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Award ID contains: 2247313

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  1. ; ; (, Communications on Pure and Applied Mathematics)
    Abstract Gromov proved a quadratic decay inequality of scalar curvature for a class of complete manifolds. In this paper, we prove that for any uniformly contractible manifold with finite asymptotic dimension, its scalar curvature decays to zero at a rate depending only on the contractibility radius of the manifold and the diameter control of the asymptotic dimension. We construct examples of uniformly contractible manifolds with finite asymptotic dimension whose scalar curvature functions decay arbitrarily slowly. This shows that our result is the best possible. We prove our result by studying the index pairing between Dirac operators and compactly supported vector bundles with Lipschitz control. A key technical ingredient for the proof of our main result is a Lipschitz control for the topologicalK‐theory of finite dimensional simplicial complexes. 
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  2. ; ; ; (, Journal of Functional Analysis)
    Free, publicly-accessible full text available January 1, 2026
  3. ; ; ; (, Annals of K-Theory)
    Free, publicly-accessible full text available December 1, 2025
  4. ; ; (, Asterisque)
    In this paper, we introduce a property of topological dynamical systems that we call finite dynamical complexity. For systems with this property, one can in principle compute the K-theory of the associated crossed product C*-algebra by splitting it up into simpler pieces and using the methods of controlled K-theory. The main part of the paper illustrates this idea by giving a new proof of the Baum-Connes conjecture for actions with finite dynamical complexity. We have tried to keep the paper as self-contained as possible: we hope the main part will be accessible to someone with the equivalent of a first course in operator K-theory. In particular, we do not assume prior knowledge of controlled K-theory, and use a new and concrete model for the Baum-Connes conjecture with coefficients that requires no bivariant K-theory to set up. 
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    Accepted Manuscript Full Text Available
  5. ; ; (, Asterisque)
    In this paper, we introduce a property of topological dynamical systems that we call finite dynamical complexity. For systems with this property, one can in principle compute the K-theory of the associated crossed product C*-algebra by splitting it up into simpler pieces and using the methods of controlled K-theory. The main part of the paper illustrates this idea by giving a new proof of the Baum-Connes conjecture for actions with finite dynamical complexity. We have tried to keep the paper as self-contained as possible: we hope the main part will be accessible to someone with the equivalent of a first course in operator K-theory. In particular, we do not assume prior knowledge of controlled K-theory, and use a new and concrete model for the Baum-Connes conjecture with coefficients that requires no bivariant K-theory to set up. 
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    Accepted Manuscript Full Text Available
  6. ; ; ; (, Mathematische Zeitschrift)
    Full Text Available 
  7. ; (, Memoirs of European Mathematical Society)
    The universal coefficient theorem is a fundamental problem in the classification theory for C*-algebras. In this article we develop a quantitative K-homology theory to prove the universal coefficient theorem for C*-algebras with finite complexity. 
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    Full Text Available