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Sökning: LAR1:gu > Chalmers tekniska högskola > Berman Robert 1976

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11.
  • Berman, Robert, 1976 (författare)
  • Bergman kernels and equilibrium measures for line bundles over projective manifolds
  • 2009
  • Ingår i: American Journal of Mathematics. - : Project Muse. - 0002-9327 .- 1080-6377. ; 131:5, s. 1485-1524
  • Tidskriftsartikel (refereegranskat)abstract
    • Let L be a holomorphic line bundle over a compact complex projective Hermitian manifold X. Any fixed smooth hermitian metric h on L induces a Hilbert space structure on the space of global holomorphic sections with values in the k th tensor power of L. In this paper various convergence results are obtained for the corresponding Bergman kernels (i.e. orthogonal projection kernels). The convergence is studied in the large k limit and is expressed in terms of the equilibrium metric h_e associated to h, as well as in terms of the Monge-Ampere measure of h on a certain support set. It is also shown that the equilibrium metric h_e is in the class C^{1,1} on the complement of the augmented base locus of L. For L ample these results give generalizations of well-known results concerning the case when the curvature of h is globally positive (then h_e=h). In general, the results can be seen as local metrized versions of Fujita's approximation theorem for the volume of L.
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12.
  • Berman, Robert, 1976 (författare)
  • Bergman kernels and equilibrium measures for polarized pseudoconcave domains
  • 2010
  • Ingår i: International Journal of Mathematics. - 0129-167X. ; 21:1, s. 77-115
  • Tidskriftsartikel (refereegranskat)abstract
    • Let X be a domain in a closed polarized complex manifold (Y, L), where L is a (semi-) positive line bundle over Y. Any given Hermitian metric on L induces by restriction to X a Hilbert space structure on the space of global holomorphic sections on Y with values in the k-th tensor power of L (also using a volume form omega(n) on X). In this paper the leading large k asymptotics for the corresponding Bergman kernels and metrics are obtained in the case when X is a pseudo-concave domain with smooth boundary (under a certain compatibility assumption). The asymptotics are expressed in terms of the curvature of L and the boundary of X. The convergence of the Bergman metrics is obtained in a more general setting where (X, omega(n)) is replaced by any measure satisfying a Bernstein-Markov property. As an application the (generalized) equilibrium measure of the polarized pseudo-concave domain X is computed explicitly. Applications to the zero and mass distribution of random holomorphic sections and the eigenvalue distribution of Toeplitz operators will be described elsewhere.
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13.
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14.
  • Berman, Robert, 1976 (författare)
  • Bergman kernels for weighted polynomials and weighted equilibrium measures of C^n
  • 2009
  • Ingår i: Indiana University Mathematics Journal. - : Indiana University Mathematics Journal. - 0022-2518. ; 58:4, s. 1921-1946
  • Tidskriftsartikel (refereegranskat)abstract
    • Various convergence results for the Bergman kernel of the Hilbert space of all polynomials in \C^{n} of total degree at most k, equipped with a weighted norm, are obtained. The weight function is assumed to be C^{1,1}, i.e. it is differentiable and all of its first partial derivatives are locally Lipshitz continuous. The convergence is studied in the large k limit and is expressed in terms of the global equilibrium potential associated to the weight function, as well as in terms of the Monge-Ampere measure of the weight function itself on a certain set. A setting of polynomials associated to a given Newton polytope, scaled by k, is also considered. These results apply directly to the study of the distribution of zeroes of random polynomials and of the eigenvalues of random normal matrices.
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15.
  • Berman, Robert, 1976, et al. (författare)
  • Convergence of Bergman measures of high powers of a line bundle
  • 2008
  • Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
    • Let L be a holomorphic line bundle on a compact complex manifold X of dimension n, and let exp(-\phi) be a continuous metric on L. Fixing a measure dμ on X gives a sequence of Hilbert spaces consisting of holomorphic sections of tensor powers of L. We prove that the corresponding sequence of scaled Bergman measures converges, in the high tensor power limit, to the equilibrium measure of the pair (K,\phi), where K is the support of dμ, as long as dμ is stably Bernstein-Markov with respect to (K,\phi). Here the Bergman measure denotes dμ times the restriction to the diagonal of the pointwise norm of the corresponding orthogonal projection operator. In particular, an extension to higher dimensions is obtained of results concerning random matrices and classical orthogonal polynomials.
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16.
  • Berman, Robert, 1976 (författare)
  • Convergence Rates for Discretized Monge-Ampere Equations and Quantitative Stability of Optimal Transport
  • 2021
  • Ingår i: Foundations of Computational Mathematics. - : Springer Science and Business Media LLC. - 1615-3375 .- 1615-3383. ; 21, s. 1099-1140
  • Tidskriftsartikel (refereegranskat)abstract
    • In recent works-both experimental and theoretical-it has been shown how to use computational geometry to efficiently construct approximations to the optimal transport map between two given probability measures on Euclidean space, by discretizing one of the measures. Here we provide a quantitative convergence analysis for the solutions of the corresponding discretized Monge-Ampere equations. This yields H-1-converge rates, in terms of the corresponding spatial resolution h, of the discrete approximations of the optimal transport map, when the source measure is discretized and the target measure has bounded convex support. Periodic variants of the results are also established. The proofs are based on new quantitative stability results for optimal transport maps, shown using complex geometry.
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17.
  • Berman, Robert, 1976, et al. (författare)
  • Convexity of the extended K-energy and the large time behavior of the weak Calabi flow
  • 2017
  • Ingår i: Geometry and Topology. - : Mathematical Sciences Publishers. - 1465-3060 .- 1364-0380. ; 21:5, s. 2945-2988
  • Tidskriftsartikel (refereegranskat)abstract
    • © 2017, Mathematical Sciences Publishers. All rights reserved. Let (X, ω) be a compact connected Kähler manifold and denote by(ε p , d p ) the metric completion of the space of Kähler potentials H ω with respect to the L p -type path length metric d p . First, we show that the natural analytic extension of the (twisted) Mabuchi K-energy to ε p is a d p -1sc functional that is convex along finite-energy geodesics. Second, following the program of J Streets, we use this to study the asymptotics of the weak (twisted) Calabi flow inside the CAT(0) metric space (ε 2 , d 2 ). This flow exists for all times and coincides with the usual smooth (twisted) Calabi flow whenever the latter exists. We show that the weak (twisted) Calabi flow either diverges with respect to the d 2 -metric or it d 1 -converges to some minimizer of the K-energy inside ε 2 . This gives the first concrete result about the long-time convergence of this flow on general Kähler manifolds, partially confirming a conjecture of Donaldson. We investigate the possibility of constructing destabilizing geodesic rays asymptotic to diverging weak (twisted) Calabi trajectories, and give a result in the case when the twisting form is Kähler. Finally, when a cscK metric exists in H ω , our results imply that the weak Calabi flow d 1 -converges to such a metric.
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18.
  • Berman, Robert, 1976, et al. (författare)
  • Convexity of the K-energy on the space of Kähler metrics and uniqueness of extremal metrics
  • 2017
  • Ingår i: Journal of the American Mathematical Society. - : American Mathematical Society (AMS). - 0894-0347 .- 1088-6834. ; 30:4, s. 1165-1196
  • Tidskriftsartikel (refereegranskat)abstract
    • We establish the convexity of Mabuchi's K-energy functional along weak geodesics in the space of Kähler potentials on a compact Kähler manifold, thus confirming a conjecture of Chen, and give some applications in Kähler geometry, including a proof of the uniqueness of constant scalar curvature metrics (or more generally extremal metrics) modulo automorphisms. The key ingredient is a new local positivity property of weak solutions to the homogeneous Monge-Ampère equation on a product domain, whose proof uses plurisubharmonic variation of Bergman kernels.
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19.
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20.
  • Berman, Robert, 1976 (författare)
  • Determinantal Point Processes and Fermions on Complex Manifolds: Large Deviations and Bosonization
  • 2014
  • Ingår i: Communications in Mathematical Physics. - : Springer Science and Business Media LLC. - 0010-3616 .- 1432-0916. ; 327:1, s. 1-47
  • Tidskriftsartikel (refereegranskat)abstract
    • We study determinantal random point processes on a compact complex manifold X associated to a Hermitian metric on a line bundle over X and a probability measure on X. Physically, this setup describes a gas of free fermions on X subject to a U(1)-gauge field and when X is the Riemann sphere it specializes to various random matrix ensembles. Our general setup will also include the setting of weighted orthogonal polynomials in , as well as in . It is shown that, in the many particle limit, the empirical random measures on X converge exponentially towards the deterministic pluripotential equilibrium measure, defined in terms of the Monge-AmpSre operator of complex pluripotential theory. More precisely, a large deviation principle (LDP) is established with a good rate functional which coincides with the (normalized) pluricomplex energy of a measure recently introduced in Berman et al. (Publ Math de l'IHA parts per thousand S 117, 179-245, 2013). We also express the LDP in terms of the Ray-Singer analytic torsion. This can be seen as an effective bosonization formula, generalizing the previously known formula in the Riemann surface case to higher dimensions and the paper is concluded with a heuristic quantum field theory interpretation of the resulting effective boson-fermion correspondence.
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  • Resultat 11-20 av 51

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