| 1. |
- Berndtsson, Bo, 1950
(författare)
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Probability measures associated to geodesics in the space of kähler metrics
- 2018
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Ingår i: Springer Proceedings in Mathematics and Statistics. - Cham : Springer International Publishing. - 2194-1017 .- 2194-1009. ; 269, s. 395-419
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Konferensbidrag (refereegranskat)abstract
- We associate certain probability measures on R to geodesics in the space HL of positively curved metrics on a line bundle L, and to geodesics in the finite dimensional symmetric space of hermitian norms on H0(X, kL). We prove that the measures associated to the finite dimensional spaces converge weakly to the measures related to geodesics in HL as k goes to infinity. The convergence of second order moments implies a recent result of Chen and Sun on geodesic distances in the respective spaces, while the convergence of first order moments gives convergence of Donaldson’s Z-functional to the Aubin–Yau energy. We also include a result on approximation of infinite dimensional geodesics by Bergman kernels which generalizes work of Phong and Sturm.
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| 2. |
- Bandara, L., et al.
(författare)
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Riesz continuity of the Atiyah-Singer Dirac operator under perturbations of the metric
- 2018
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Ingår i: Mathematische Annalen. - : Springer Science and Business Media LLC. - 0025-5831 .- 1432-1807. ; 370:1-2, s. 863-915
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Tidskriftsartikel (refereegranskat)abstract
- We prove that the Atiyah-Singer Dirac operator D-g in L-2 depends Riesz continuously L-infinity on perturbations of complete metrics on a smooth manifold. The Lipschitz bound for the map g -> D-g(1 + D-g(2))(-1/2) depends on bounds on Ricci curvature and its first derivatives as well as a lower bound on injectivity radius. Our proof uses harmonic analysis techniques related to Caldern's first commutator and the Kato square root problem. We also show perturbation results for more general functions of general Dirac-type operators on vector bundles.
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| 3. |
- Bogfjellmo, Geir, 1987, et al.
(författare)
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A Numerical Algorithm for C-2-Splines on Symmetric Spaces
- 2018
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Ingår i: SIAM Journal on Numerical Analysis. - : Siam Publications. - 1095-7170 .- 0036-1429. ; 56:4, s. 2623-2647
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Tidskriftsartikel (refereegranskat)abstract
- Cubic spline interpolation on Euclidean space is a standard topic in numerical analysis, with countless applications in science and technology. In several emerging fields, for example, computer vision and quantum control, there is a growing need for spline interpolation on curved, non-Euclidean space. The generalization of cubic splines to manifolds is not self-evident, with several distinct approaches. One possibility is to mimic the acceleration minimizing property, which leads to Riemannian cubics. This, however, requires the solution of a coupled set of nonlinear boundary value problems that cannot be integrated explicitly, even if formulae for geodesics are available. Another possibility is to mimic De Casteljau's algorithm, which leads to generalized .Bezier curves. To construct C-2-splines from such curves is a complicated nonlinear problem, until now lacking numerical methods. Here we provide an iterative algorithm for C-2-splines on Riemannian symmetric spaces, and we prove convergence of linear order. In terms of numerical tractability and computational efficiency, the new method surpasses those based on Riemannian cubics. Each iteration is parallel and thus suitable for multicore implementation. We demonstrate the algorithm for three geometries of interest: the n-sphere, complex projective space, and the real Grassmannian.
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| 4. |
- Albin, Patrik, 1960
(författare)
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ON EXTREME VALUE THEORY FOR GROUP STATIONARY GAUSSIAN PROCESSES
- 2018
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Ingår i: Esaim-Probability and Statistics. - : EDP Sciences. - 1292-8100 .- 1262-3318. ; 22, s. 1-18
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Tidskriftsartikel (refereegranskat)abstract
- We study extreme value theory of right stationary Gaussian processes with parameters in open subsets with compact closure of (not necessarily Abelian) locally compact topological groups. Even when specialized to Euclidian space our result extend results on extremes of stationary Gaussian processes and fields in the literature by means of requiring weaker technical conditions as well as by means of the fact that group stationary processes need not be stationary in the usual sense (that is, with respect to addition as group operation).
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| 5. |
- Aldana, C. L., et al.
(författare)
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A Polyakov Formula for Sectors
- 2018
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Ingår i: Journal of Geometric Analysis. - : Springer Science and Business Media LLC. - 1050-6926 .- 1559-002X. ; 28:2, s. 1773-1839
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Tidskriftsartikel (refereegranskat)abstract
- We consider finite area convex Euclidean circular sectors. We prove a variational Polyakov formula which shows how the zeta-regularized determinant of the Laplacian varies with respect to the opening angle. Varying the angle corresponds to a conformal deformation in the direction of a conformal factor with a logarithmic singularity at the origin. We compute explicitly all the contributions to this formula coming from the different parts of the sector. In the process, we obtain an explicit expression for the heat kernel on an infinite area sector using Carslaw-Sommerfeld's heat kernel. We also compute the zeta-regularized determinant of rectangular domains of unit area and prove that it is uniquely maximized by the square.
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| 6. |
- Andersson, Mats, 1957
(författare)
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A global Briançon-Skoda-Huneke-Sznajdman theorem
- 2018
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Ingår i: Mathematica Scandinavica. - : Det Kgl. Bibliotek/Royal Danish Library. - 1903-1807 .- 0025-5521. ; 122:1, s. 31-52
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Tidskriftsartikel (refereegranskat)abstract
- We prove a global effective membership result for polynomials on a non-reduced algebraic sub variety of C-N. It can be seen as a global version of a recent local result of Sznajdman. generalizine the Briancon-Skoda-Huneke theorem for the local ring of holomorphic functions at a point on a reduced analytic space.
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| 7. |
- Brandes, Julia, 1986
(författare)
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On the number of linear spaces on hypersurfaces with a prescribed discriminant
- 2018
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Ingår i: Mathematische Zeitschrift. - : Springer Science and Business Media LLC. - 1432-1823 .- 0025-5874. ; 289:3-4, s. 803-827
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Tidskriftsartikel (refereegranskat)abstract
- For a given form we apply the circle method in order to give an asymptotic estimate of the number of m-tuples spanning a linear space on the hypersurface with the property that . This allows us in some measure to count rational linear spaces on hypersurfaces whose underlying integer lattice is primitive.
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| 8. |
- 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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| 9. |
- Eriksson, Dennis, 1981, et al.
(författare)
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Singularities of metrics on Hodge bundles and their topological invariants
- 2018
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Ingår i: Algebraic Geometry. - 2313-1691 .- 2214-2584. ; 5:6, s. 742-775
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Tidskriftsartikel (refereegranskat)abstract
- We consider degenerations of complex projective Calabi-Yau varieties and study the singularities of L2, Quillen and BCOV metrics on Hodge and determinant bundles. The dominant and subdominant terms in the expansions of the metrics close to non-smooth fibers are shown to be related to well-known topological invariants of singularities, such as limit Hodge structures, vanishing cycles and log-canonical thresholds. We also describe corresponding invariants for more general degenerating families in the case of the Quillen metric.
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| 10. |
- Hulek, Klaus, et al.
(författare)
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The topology of Ag and Its Compactifications
- 2018
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Ingår i: Abel Symposia. - Cham : Springer International Publishing. - 2193-2808 .- 2197-8549. ; 14, s. 135-193
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Konferensbidrag (refereegranskat)abstract
- We survey old and new results about the cohomology of the moduli space Ag of principally polarized abelian varieties of genus g and its compactifications. The main emphasis lies on the computation of the cohomology for small genus and on stabilization results. We review both geometric and representation theoretic approaches to the problem. The appendix provides a detailed discussion of computational methods based on trace formulae and automorphic representations, in particular Arthur’s endoscopic classification of automorphic representations for symplectic groups.
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