Random polynomials and random Kähler geometry


Description Phase 2

The research in this proposal focuses on the interplay between complex geometry and probability theory. More specifically, we aim to combine methods from complex geometry and geometric analysis with probabilistic techniques to study several problems concerning local and global statistical properties of zeros of holomorphic sections of holomorphic line bundles over Kähler manifolds. A particularly important example of this setting is the case of random polynomials. We are interested in the asymptotics of the covariance kernels of the polynomial/section ensembles, the universality of their distributions, central limit theorems and large deviation principles. On the one hand, the questions we plan to investigate are interesting and challenging from a mathematical point of view. On the other hand, in recent decades they have also shown important connections to theoretical physics, where random polynomials serve as a basic model for the eigenfunctions of quantum chaotic Hamiltonians.

Description Phase 1

The research in this proposal focuses on the interplay between complex geometry and probability theory. More precisely, we aim to combine methods from complex geometry and geometric analysis with probabilistic techniques in order to study several problems concerning local and global statistical properties of zeros of holomorphic sections of holomorphic line bundles over Kähler manifolds. A particularly important instance of this setting is given by the case of random polynomials. We are interested in the asymptotics of the covariance kernels of the polynomial / sections ensembles, universality of their distributions, central limit theorems and large deviation principles in this context.

On the one hand, the questions we are planning to investigate are interesting and challenging from a mathematical point of view. On the other hand, during the last couple of decades, they have also been exhibiting important connections to theoretical physics, where random polynomials serve as a basic model for eigenfunctions of quantum chaotic Hamiltonians.

Preprints/Publications

Alexander Drewitz, Bingxiao Liu, George Marinescu: Gaussian holomorphic sections on noncompact complex manifolds (02/2023)

Alexander Drewitz, Bingxiao Liu, George Marinescu: Large deviations for zeros of holomorphic sections on punctured Riemann surfaces (09/2021)

Dan Coman, George Marinescu: Equidistribution for weakly holomorphic sections of line bundles on algebraic curves (12/2020) published

Dan Coman, Wen Lu, Ma Xiaonan, George Marinescu: Bergman kernels and equidistribution for sequences of line bundles on Kähler manifolds (12/2020)

Members

  • member's portrait

    Prof. Dr. Alexander Drewitz

    Universität zu Köln
    Principal Investigator
  • member's portrait

    Prof. Dr. George Marinescu

    Universität zu Köln
    Principal Investigator
  • member's portrait

    M. Sc. Paul Klass

    Universität zu Köln
    Associated Scientist
  • member's portrait

    Dr. Bingxiao Liu

    Universität zu Köln
    Associated Scientist

Project Related News