Project description
New tools for inverse problems of spectral shape analysis
In 1911, Hermann Weyl showed that the volume of a bounded domain in the Euclidean space is determined by the asymptotic behaviour of the eigenvalues of the Dirichlet Laplacian operator. The work motivated the title of a famous paper by M. Kac: 'Can you hear the shape of a drum?' The question is whether the shape of a bounded domain can be determined by the spectrum of the Dirichlet Laplacian operator. The existence of isospectral domains (domains with the same number of eigenvalues) which do not preserve the shape of planes during transformations (non-isometric) remains an open question. The goal of the EU-funded SPERIG project is to develop tools to solve the local inverse problems for smooth planar convex domains and geodesic flows.
Objective
In 1911, Hermann Weyl proved the remarkable asymptotic formula describing distribution of (large) eigenvalues of the Dirichlet Laplacian in a bounded domain Ω ⊂ Rd
N (λ) = (2π)−d ωd Vol(Ω) λd/2(1 + o(1)) as λ → +∞. where N (λ) is the number of eigenvalues of the Laplacian spectrum, which are less than λ, ωd is a volume of the unit ball in Rd, Vol(Ω) is the volume of Ω, and the Laplace spectrum of a domain Ω is defined as the set of positive real numbers λ (with multiplicities) that satisfy the eigenvalue problem in Ω with Dirichlet boundary conditions. This result motivated the title of a famous paper by M. Kac “Can you hear the shape of a drum?”. The question is: can the shape of a bounded domain O C Rd be determined by the Laplace spectrum? Two domains are called isospectral if they have the same eigenvalues. Consider the space of domains with a smooth boundary. The existence of isospectral non-isometric domains is a well-known open question.
The first goal of the project is to prove the local spectral rigidity for convex planar domains, i.e. for a smooth convex planar domain Ω the Laplace spectrum determines Ω locally. There are no nearby isospectral non-isometric domains with smooth boundary. All of the these questions can also be posed for Riemannian manifolds. The second goal is to prove the local rigidity for Riemannian manifolds with Anosov geodesic flows.
The third goal is to prove local rigidity for integrable systems: geodesic flows on tori (resp. convex planar billiards). The goal is to prove that an integrable metric close to a Liouville metric is Liouville. The second type is billiards inside smooth planar domains integrable near the boundary. We shall prove that domains with integrable billiards belong to a finite-dimensional manifold.
The focal goal of the project is to develop analytic tools to solve the local inverse problems for smooth planar convex domains and geodesic flows.
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Project’s keywords as indicated by the project coordinator. Not to be confused with the EuroSciVoc taxonomy (Fields of science)
Programme(s)
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Multi-annual funding programmes that define the EU’s priorities for research and innovation.
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H2020-EU.1.1. - EXCELLENT SCIENCE - European Research Council (ERC)
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Calls for proposals are divided into topics. A topic defines a specific subject or area for which applicants can submit proposals. The description of a topic comprises its specific scope and the expected impact of the funded project.
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ERC-ADG - Advanced Grant
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Call for proposal
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(opens in new window) ERC-2019-ADG
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3400 Klosterneuburg
Austria
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