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Colloquium
Seminar information archive ～10/03｜Next seminar｜Future seminars 10/04～
| Organizer(s) | AIDA Shigeki (chair), IKE Yuichi, IMAI Naoki, HAYASHI Shuhei |
|---|---|
| URL | https://www.ms.u-tokyo.ac.jp/seminar/colloquium_e/index_e.html |
2026/10/16
15:30-16:30 Room #NISSAY Lecture Hall (Graduate School of Math. Sci. Bldg.)
Kouichi Taira (Graduate School of Mathematical Sciences, The University of Tokyo)
Spectral theory for the d'Alembertian on Lorentzian manifolds (日本語)
Kouichi Taira (Graduate School of Mathematical Sciences, The University of Tokyo)
Spectral theory for the d'Alembertian on Lorentzian manifolds (日本語)
[ Abstract ]
The spectrum of a linear operator on a Hilbert space may be regarded as an infinite-dimensional generalization of the notion of eigenvalues of matrices, and it has been studied extensively from the viewpoints of analysis and geometry. For example, the spectra of the Laplacian and Dirac operators on Riemannian manifolds are closely related to a variety of geometric phenomena.
Classical results include Weyl's law, which describes the asymptotic behavior of eigenvalues in terms of geometric quantities such as the volume of the manifold, and the index theorem, which relates the zero modes of the Dirac operator to characteristic classes of the manifold. These topics continue to be actively studied today.
On Lorentzian manifolds, which provide the geometric framework for relativity, the spectra of operators such as the d'Alembertian and the Dirac operator have also begun to attract increasing attention in recent years, partly because of applications to quantum theory. In contrast to the Riemannian setting, however, these differential operators are not elliptic, and many of the standard methods are therefore no longer directly applicable. As a result, even fundamental questions such as self-adjointness can be highly nontrivial.
In this talk, I will give an overview of known results on the spectra of differential operators on Riemannian and Lorentzian manifolds and, time permitting, present some of my own results.
The spectrum of a linear operator on a Hilbert space may be regarded as an infinite-dimensional generalization of the notion of eigenvalues of matrices, and it has been studied extensively from the viewpoints of analysis and geometry. For example, the spectra of the Laplacian and Dirac operators on Riemannian manifolds are closely related to a variety of geometric phenomena.
Classical results include Weyl's law, which describes the asymptotic behavior of eigenvalues in terms of geometric quantities such as the volume of the manifold, and the index theorem, which relates the zero modes of the Dirac operator to characteristic classes of the manifold. These topics continue to be actively studied today.
On Lorentzian manifolds, which provide the geometric framework for relativity, the spectra of operators such as the d'Alembertian and the Dirac operator have also begun to attract increasing attention in recent years, partly because of applications to quantum theory. In contrast to the Riemannian setting, however, these differential operators are not elliptic, and many of the standard methods are therefore no longer directly applicable. As a result, even fundamental questions such as self-adjointness can be highly nontrivial.
In this talk, I will give an overview of known results on the spectra of differential operators on Riemannian and Lorentzian manifolds and, time permitting, present some of my own results.
