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Article Dans Une Revue Asian Journal of Mathematics Année : 2017

Betti numbers of random nodal sets of elliptic pseudo-differential operators

Damien Gayet

Résumé

Given an elliptic self-adjoint pseudo-differential operator $P$ bounded from below, acting on the sections of a Riemannian line bundle over a smooth closed manifold $M$ equipped with some Lebesgue measure, we estimate from above, as $L$ grows to infinity, the Betti numbers of the vanishing locus of a random section taken in the direct sum of the eigenspaces of $P$ with eigenvalues below $L$. These upper estimates follow from some equidistribution of the critical points of the restriction of a fixed Morse function to this vanishing locus. We then consider the examples of the Laplace-Beltrami and the Dirichlet-to-Neumann operators associated to some Riemannian metric on $M$.
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Dates et versions

hal-00999485 , version 1 (03-06-2014)

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Damien Gayet, Jean-Yves Welschinger. Betti numbers of random nodal sets of elliptic pseudo-differential operators. Asian Journal of Mathematics, 2017, 21 (5), pp.811-840. ⟨10.4310/AJM.2017.v21.n5.a2⟩. ⟨hal-00999485⟩
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