| 1. |
- Gimperlein, H., et al.
(författare)
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On the magnitude function of domains in Euclidean space
- 2021
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Ingår i: American Journal of Mathematics. - : Project Muse. - 0002-9327 .- 1080-6377. ; 143:3, s. 939-967
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Tidskriftsartikel (refereegranskat)abstract
- We study Leinster's notion of magnitude for a compact metric space. For a smooth, compact domain X subset of R2m-1, we find geometric significance in the function M-X(R) = mag(R . X). The function M-X extends from the positive half-line to a meromorphic function in the complex plane. Its poles are generalized scattering resonances. In the semiclassical limit R -> infinity, M-X admits an asymptotic expansion. The three leading terms of M-X at R = +infinity are proportional to the volume, surface area and integral of the mean curvature. In particular, for convex X the leading terms are proportional to the intrinsic volumes, and we obtain an asymptotic variant of the convex magnitude conjecture by Leinster and Willerton, with corrected coefficients.
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| 2. |
- Deeley, R.J., et al.
(författare)
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Applying geometric K-cycles to fractional indices
- 2017
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Ingår i: Mathematische Nachrichten. - : Wiley. - 1522-2616 .- 0025-584X. ; 290:14-15, s. 2207-2233
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Tidskriftsartikel (refereegranskat)abstract
- A geometric model for twisted K-homology is introduced. It is modeled after the Mathai–Melrose–Singer fractional analytic index theorem in the same way as the Baum–Douglas model of K-homology was modeled after the Atiyah–Singer index theorem. A natural transformation from twisted geometric K-homology to the new geometric model is constructed. The analytic assembly mapping to analytic twisted K-homology in this model is an isomorphism for torsion twists on a finite CW-complex. For a general twist on a smooth manifold the analytic assembly mapping is a surjection. Beyond the aforementioned fractional invariants, we study T-duality for geometric cycles.
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| 3. |
- Deeley, R.J., et al.
(författare)
-
Realizing the analytic surgery group of Higson and Roe geometrically, part I: the geometric model
- 2017
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Ingår i: Journal of Homotopy and Related Structures. - : Springer Science and Business Media LLC. - 2193-8407 .- 1512-2891. ; 12:1, s. 109-142
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Tidskriftsartikel (refereegranskat)abstract
- We construct a geometric analog of the analytic surgery group of Higson and Roe for the assembly mapping for free actions of a group with values in a Banach algebra completion of the group algebra. We prove that the geometrically defined group, in analogy with the analytic surgery group, fits into a six term exact sequence with the assembly mapping and also discuss mappings with domain the geometric group. In particular, given two finite dimensional unitary representations of the same rank, we define a map in the spirit of η-type invariants from the geometric group (with respect to assembly for the full group C ∗ -algebra) to the real numbers.
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| 4. |
- Deeley, R.J., et al.
(författare)
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Realizing the analytic surgery group of Higson and Roe geometrically part II: relative η -invariants
- 2016
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Ingår i: Mathematische Annalen. - : Springer Science and Business Media LLC. - 0025-5831 .- 1432-1807. ; 366:3-4, s. 1319-1363
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Tidskriftsartikel (refereegranskat)abstract
- We apply the geometric analog of the analytic surgery group of Higson and Roe to the relative η-invariant. In particular, by solving a Baum–Douglas type index problem, we give a “geometric” proof of a result of Keswani regarding the homotopy invariance of relative η-invariants. The starting point for this work is our previous constructions in “Realizing the analytic surgery group of Higson and Roe geometrically, Part I: The geometric model”.
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| 5. |
- Deeley, R.J., et al.
(författare)
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Realizing the analytic surgery group of Higson and Roe geometrically part III: higher invariants
- 2016
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Ingår i: Mathematische Annalen. - : Springer Science and Business Media LLC. - 0025-5831 .- 1432-1807. ; 366:3-4, s. 1513-1559
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Tidskriftsartikel (refereegranskat)abstract
- We construct an isomorphism between the geometric model and Higson-Roe’s analytic surgery group, reconciling the constructions in the previous papers in the series on “Realizing the analytic surgery group of Higson and Roe geometrically” with their analytic counterparts. Following work of Lott and Wahl, we construct a Chern character on the geometric model for the surgery group; it is a “delocalized Chern character”, from which Lott’s higher delocalized ρ-invariants can be retrieved. Following work of Piazza and Schick, we construct a geometric map from Stolz’ positive scalar curvature sequence to the geometric model of Higson-Roe’s analytic surgery exact sequence.
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| 6. |
- Deeley, Robin J., et al.
(författare)
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Relative geometric assembly and mapping cones, part I: the geometric model and applications
- 2018
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Ingår i: Journal of Topology. - : Wiley. - 1753-8416 .- 1753-8424. ; 11:4, s. 966-1000
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Tidskriftsartikel (refereegranskat)abstract
- 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 (for example, the localized index and higher Atiyah–Patodi–Singer index theory). As an application we obtain a vanishing result in the context of manifolds with boundary and positive scalar curvature; this result is also inspired and connected to the work of Chang, Weinberger and Yu. Furthermore, we use results of Wahl to show that rational injectivity of the relative assembly map implies homotopy invariance of the relative higher signatures of a manifold with boundary.
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| 7. |
- Deeley, Robin J., et al.
(författare)
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Relative geometric assembly and mapping cones Part II: Chern characters and the Novikov property
- 2019
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Ingår i: Münster Journal of Mathematics. - 1867-5778 .- 1867-5786. ; 12:1, s. 57-92
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Tidskriftsartikel (refereegranskat)abstract
- We study Chern characters and the assembly mapping for free actions using the framework of geometric K-homology. The focus is on the relative groups associated with a group homomorphism phi: Gamma(1) -> Gamma(2) along with applications to Novikov type properties. In particular, we prove a relative strong Novikov property for homomorphisms of hyperbolic groups and a relative strong l(1)-Novikov property for polynomially bounded homomorphisms of groups with polynomially bounded cohomology in C. As a corollary, relative higher signatures on a manifold with boundary W, with pi(1)(partial derivative W) -> pi(1) (W) belonging to the class above, are homotopy invariant.
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| 8. |
- Deeley, R. J., et al.
(författare)
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SMALE SPACE C*-ALGEBRAS HAVE NONZERO PROJECTIONS
- 2020
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Ingår i: Proceedings of the American Mathematical Society. - : American Mathematical Society (AMS). - 0002-9939 .- 1088-6826. ; 148:4, s. 1625-1639
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Tidskriftsartikel (refereegranskat)abstract
- The main result of the present paper is that the stable and unstable C*-algebras associated to a mixing Smale space always contain nonzero projections. This gives a positive answer to a question of the first listed author and Karen Strung and has implications for the structure of these algebras in light of the Elliott program for simple C*-algebras. Using our main result, we also show that the homoclinic, stable, and unstable algebras each have real rank zero.
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| 9. |
- Deeley, Robin J., et al.
(författare)
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The bordism group of unbounded KK-cycles
- 2018
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Ingår i: Journal of Topology and Analysis. - 1793-5253 .- 1793-7167. ; 10:2
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Tidskriftsartikel (refereegranskat)abstract
- © 2018 World Scientific Publishing Company We consider Hilsum’s notion of bordism as an equivalence relation on unbounded (Formula presented.)-cycles and study the equivalence classes. Upon fixing two (Formula presented.)-algebras, and a ∗-subalgebra dense in the first (Formula presented.)-algebra, a (Formula presented.)-graded abelian group is obtained; it maps to the Kasparov (Formula presented.)-group of the two (Formula presented.)-algebras via the bounded transform. We study properties of this map both in general and in specific examples. In particular, it is an isomorphism if the first (Formula presented.)-algebra is the complex numbers (i.e. for (Formula presented.)-theory) and is a split surjection if the first (Formula presented.)-algebra is the continuous functions on a compact manifold with boundary when one uses the Lipschitz functions as the dense ∗-subalgebra.
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| 10. |
- DEELEY, ROBIN J., et al.
(författare)
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Wieler solenoids, Cuntz–Pimsner algebras and K-theory
- 2018
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Ingår i: Ergodic Theory and Dynamical Systems. - : Cambridge University Press (CUP). - 0143-3857 .- 1469-4417. ; 38:8, s. 2942-2988
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Tidskriftsartikel (refereegranskat)abstract
- © Cambridge University Press, 2017 We study irreducible Smale spaces with totally disconnected stable sets and their associated (Formula presented.)-theoretic invariants. Such Smale spaces arise as Wieler solenoids, and we restrict to those arising from open surjections. The paper follows three converging tracks: one dynamical, one operator algebraic and one (Formula presented.)-theoretic. Using Wieler’s theorem, we characterize the unstable set of a finite set of periodic points as a locally trivial fibre bundle with discrete fibres over a compact space. This characterization gives us the tools to analyse an explicit groupoid Morita equivalence between the groupoids of Deaconu–Renault and Putnam–Spielberg, extending results of Thomsen. The Deaconu–Renault groupoid and the explicit Morita equivalence lead to a Cuntz–Pimsner model for the stable Ruelle algebra. The (Formula presented.)-theoretic invariants of Cuntz–Pimsner algebras are then studied using the Cuntz–Pimsner extension, for which we construct an unbounded representative. To elucidate the power of these constructions, we characterize the Kubo–Martin–Schwinger (KMS) weights on the stable Ruelle algebra of a Wieler solenoid. We conclude with several examples of Wieler solenoids, their associated algebras and spectral triples.
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