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Can you hear the shape of a drum ? and Deformational Spectral Rigidity

• V. Kaloshin

February 7, 2019

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 1 / 28

Can you hear the shape of a drum? and Deformational Spectral Rigidity

• M. Kac ‘Can you hear the shape of a drum?’

Laplace spectrum, Inverse problems Length spectrum and Laplace spectrum Deformational Spectral Rigidity and Main Results Ideas of proofs

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 2 / 28

Can you hear the shape of a drum? and Deformational Spectral Rigidity

• M. Kac ‘Can you hear the shape of a drum?’

Laplace spectrum, Inverse problems Length spectrum and Laplace spectrum Deformational Spectral Rigidity and Main Results Ideas of proofs

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 2 / 28

Can you hear the shape of a drum? and Deformational Spectral Rigidity

• M. Kac ‘Can you hear the shape of a drum?’

Laplace spectrum, Inverse problems Length spectrum and Laplace spectrum Deformational Spectral Rigidity and Main Results Ideas of proofs

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 2 / 28

Can you hear the shape of a drum? and Deformational Spectral Rigidity

• M. Kac ‘Can you hear the shape of a drum?’

Laplace spectrum, Inverse problems Length spectrum and Laplace spectrum Deformational Spectral Rigidity and Main Results Ideas of proofs

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 2 / 28

Can you hear the shape of a drum? and Deformational Spectral Rigidity

• M. Kac ‘Can you hear the shape of a drum?’

Laplace spectrum, Inverse problems Length spectrum and Laplace spectrum Deformational Spectral Rigidity and Main Results Ideas of proofs

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 2 / 28

Can you hear the shape of a drum? and Deformational Spectral Rigidity

• M. Kac ‘Can you hear the shape of a drum?’

Laplace spectrum, Inverse problems Length spectrum and Laplace spectrum Deformational Spectral Rigidity and Main Results Ideas of proofs

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 2 / 28

Can you hear the shape of a drum? and Deformational Spectral Rigidity

• M. Kac ‘Can you hear the shape of a drum?’

Laplace spectrum, Inverse problems Length spectrum and Laplace spectrum Deformational Spectral Rigidity and Main Results Ideas of proofs

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 2 / 28

Can you hear the shape of a drum? and Deformational Spectral Rigidity

• M. Kac ‘Can you hear the shape of a drum?’

Laplace spectrum, Inverse problems Length spectrum and Laplace spectrum Deformational Spectral Rigidity and Main Results Ideas of proofs

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 2 / 28

Can you hear the shape of a drum? and Deformational Spectral Rigidity

• M. Kac ‘Can you hear the shape of a drum?’

Laplace spectrum, Inverse problems Length spectrum and Laplace spectrum Deformational Spectral Rigidity and Main Results Ideas of proofs

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 2 / 28

Can you hear the shape of a drum?

• M. Kac’66: Can you hear the shape of a drum?
• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 3 / 28

Can you hear the shape of a drum?

Consider the Dirichlet problem in a domain Ω ⊂ R2.

• ∆u + λ2u = 0

u|∂Ω = 0. ∆(Ω) := {0 < λ1 ≤ λ2 ≤ · · · } — Laplace spectrum. Example 1 Let ΩC = [0, π] × [0, π] ∋ (x, y). For any pair k, m ∈ Z+ \ 0 let u(x, y) = sin kx · sin my and λ =

• k2 + m2.

The Laplace spectrum ∆(ΩC) = ∪k,m∈Z+\0 √ k2 + m2. Question (M. Kac’66) Does ∆(Ω) determine Ω up to isometry? Weyl law (H. Weyl’11) N(λ) := # eigenvalues (w multiplicity) in (0, λ2], then lim

λ→∞ λ−1N(λ) = (4π)−1Area (Ω).

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 4 / 28

Can you hear the shape of a drum?

Consider the Dirichlet problem in a domain Ω ⊂ R2.

• ∆u + λ2u = 0

u|∂Ω = 0. ∆(Ω) := {0 < λ1 ≤ λ2 ≤ · · · } — Laplace spectrum. Example 1 Let ΩC = [0, π] × [0, π] ∋ (x, y). For any pair k, m ∈ Z+ \ 0 let u(x, y) = sin kx · sin my and λ =

• k2 + m2.

The Laplace spectrum ∆(ΩC) = ∪k,m∈Z+\0 √ k2 + m2. Question (M. Kac’66) Does ∆(Ω) determine Ω up to isometry? Weyl law (H. Weyl’11) N(λ) := # eigenvalues (w multiplicity) in (0, λ2], then lim

λ→∞ λ−1N(λ) = (4π)−1Area (Ω).

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 4 / 28

Can you hear the shape of a drum?

Consider the Dirichlet problem in a domain Ω ⊂ R2.

• ∆u + λ2u = 0

u|∂Ω = 0. ∆(Ω) := {0 < λ1 ≤ λ2 ≤ · · · } — Laplace spectrum. Example 1 Let ΩC = [0, π] × [0, π] ∋ (x, y). For any pair k, m ∈ Z+ \ 0 let u(x, y) = sin kx · sin my and λ =

• k2 + m2.

The Laplace spectrum ∆(ΩC) = ∪k,m∈Z+\0 √ k2 + m2. Question (M. Kac’66) Does ∆(Ω) determine Ω up to isometry? Weyl law (H. Weyl’11) N(λ) := # eigenvalues (w multiplicity) in (0, λ2], then lim

λ→∞ λ−1N(λ) = (4π)−1Area (Ω).

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 4 / 28

Can you hear the shape of a drum?

Consider the Dirichlet problem in a domain Ω ⊂ R2.

• ∆u + λ2u = 0

u|∂Ω = 0. ∆(Ω) := {0 < λ1 ≤ λ2 ≤ · · · } — Laplace spectrum. Example 1 Let ΩC = [0, π] × [0, π] ∋ (x, y). For any pair k, m ∈ Z+ \ 0 let u(x, y) = sin kx · sin my and λ =

• k2 + m2.

The Laplace spectrum ∆(ΩC) = ∪k,m∈Z+\0 √ k2 + m2. Question (M. Kac’66) Does ∆(Ω) determine Ω up to isometry? Weyl law (H. Weyl’11) N(λ) := # eigenvalues (w multiplicity) in (0, λ2], then lim

λ→∞ λ−1N(λ) = (4π)−1Area (Ω).

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 4 / 28

Can you hear the shape of a drum?

Consider the Dirichlet problem in a domain Ω ⊂ R2.

• ∆u + λ2u = 0

u|∂Ω = 0. ∆(Ω) := {0 < λ1 ≤ λ2 ≤ · · · } — Laplace spectrum. Example 1 Let ΩC = [0, π] × [0, π] ∋ (x, y). For any pair k, m ∈ Z+ \ 0 let u(x, y) = sin kx · sin my and λ =

• k2 + m2.

The Laplace spectrum ∆(ΩC) = ∪k,m∈Z+\0 √ k2 + m2. Question (M. Kac’66) Does ∆(Ω) determine Ω up to isometry? Weyl law (H. Weyl’11) N(λ) := # eigenvalues (w multiplicity) in (0, λ2], then lim

λ→∞ λ−1N(λ) = (4π)−1Area (Ω).

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 4 / 28

Can you hear the shape of a drum?

Consider the Dirichlet problem in a domain Ω ⊂ R2.

• ∆u + λ2u = 0

u|∂Ω = 0. ∆(Ω) := {0 < λ1 ≤ λ2 ≤ · · · } — Laplace spectrum. Example 1 Let ΩC = [0, π] × [0, π] ∋ (x, y). For any pair k, m ∈ Z+ \ 0 let u(x, y) = sin kx · sin my and λ =

• k2 + m2.

The Laplace spectrum ∆(ΩC) = ∪k,m∈Z+\0 √ k2 + m2. Question (M. Kac’66) Does ∆(Ω) determine Ω up to isometry? Weyl law (H. Weyl’11) N(λ) := # eigenvalues (w multiplicity) in (0, λ2], then lim

λ→∞ λ−1N(λ) = (4π)−1Area (Ω).

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 4 / 28

Can you hear the shape of a drum?

Consider the Dirichlet problem in a domain Ω ⊂ R2.

• ∆u + λ2u = 0

u|∂Ω = 0. ∆(Ω) := {0 < λ1 ≤ λ2 ≤ · · · } — Laplace spectrum. Example 1 Let ΩC = [0, π] × [0, π] ∋ (x, y). For any pair k, m ∈ Z+ \ 0 let u(x, y) = sin kx · sin my and λ =

• k2 + m2.

The Laplace spectrum ∆(ΩC) = ∪k,m∈Z+\0 √ k2 + m2. Question (M. Kac’66) Does ∆(Ω) determine Ω up to isometry? Weyl law (H. Weyl’11) N(λ) := # eigenvalues (w multiplicity) in (0, λ2], then lim

λ→∞ λ−1N(λ) = (4π)−1Area (Ω).

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 4 / 28

Can you hear the shape of a drum?

Consider the Dirichlet problem in a domain Ω ⊂ R2.

• ∆u + λ2u = 0

u|∂Ω = 0. ∆(Ω) := {0 < λ1 ≤ λ2 ≤ · · · } — Laplace spectrum. Example 1 Let ΩC = [0, π] × [0, π] ∋ (x, y). For any pair k, m ∈ Z+ \ 0 let u(x, y) = sin kx · sin my and λ =

• k2 + m2.

The Laplace spectrum ∆(ΩC) = ∪k,m∈Z+\0 √ k2 + m2. Question (M. Kac’66) Does ∆(Ω) determine Ω up to isometry? Weyl law (H. Weyl’11) N(λ) := # eigenvalues (w multiplicity) in (0, λ2], then lim

λ→∞ λ−1N(λ) = (4π)−1Area (Ω).

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 4 / 28

Can you hear the shape of a drum?

Consider the Dirichlet problem in a domain Ω ⊂ R2.

• ∆u + λ2u = 0

u|∂Ω = 0. ∆(Ω) := {0 < λ1 ≤ λ2 ≤ · · · } — Laplace spectrum. Example 1 Let ΩC = [0, π] × [0, π] ∋ (x, y). For any pair k, m ∈ Z+ \ 0 let u(x, y) = sin kx · sin my and λ =

• k2 + m2.

The Laplace spectrum ∆(ΩC) = ∪k,m∈Z+\0 √ k2 + m2. Question (M. Kac’66) Does ∆(Ω) determine Ω up to isometry? Weyl law (H. Weyl’11) N(λ) := # eigenvalues (w multiplicity) in (0, λ2], then lim

λ→∞ λ−1N(λ) = (4π)−1Area (Ω).

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 4 / 28

Can you hear the shape of a drum?

Consider the Dirichlet problem in a domain Ω ⊂ R2.

• ∆u + λ2u = 0

u|∂Ω = 0. ∆(Ω) := {0 < λ1 ≤ λ2 ≤ · · · } — Laplace spectrum. Example 1 Let ΩC = [0, π] × [0, π] ∋ (x, y). For any pair k, m ∈ Z+ \ 0 let u(x, y) = sin kx · sin my and λ =

• k2 + m2.

The Laplace spectrum ∆(ΩC) = ∪k,m∈Z+\0 √ k2 + m2. Question (M. Kac’66) Does ∆(Ω) determine Ω up to isometry? Weyl law (H. Weyl’11) N(λ) := # eigenvalues (w multiplicity) in (0, λ2], then lim

λ→∞ λ−1N(λ) = (4π)−1Area (Ω).

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 4 / 28

Can you hear the shape of a drum?

Consider the Dirichlet problem in a domain Ω ⊂ R2.

• ∆u + λ2u = 0

u|∂Ω = 0. ∆(Ω) := {0 < λ1 ≤ λ2 ≤ · · · } — Laplace spectrum. Example 1 Let ΩC = [0, π] × [0, π] ∋ (x, y). For any pair k, m ∈ Z+ \ 0 let u(x, y) = sin kx · sin my and λ =

• k2 + m2.

The Laplace spectrum ∆(ΩC) = ∪k,m∈Z+\0 √ k2 + m2. Question (M. Kac’66) Does ∆(Ω) determine Ω up to isometry? Weyl law (H. Weyl’11) N(λ) := # eigenvalues (w multiplicity) in (0, λ2], then lim

λ→∞ λ−1N(λ) = (4π)−1Area (Ω).

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 4 / 28

Can you hear the shape of a drum?

Consider the Dirichlet problem in a domain Ω ⊂ R2.

• ∆u + λ2u = 0

u|∂Ω = 0. ∆(Ω) := {0 < λ1 ≤ λ2 ≤ · · · } — Laplace spectrum. Example 1 Let ΩC = [0, π] × [0, π] ∋ (x, y). For any pair k, m ∈ Z+ \ 0 let u(x, y) = sin kx · sin my and λ =

• k2 + m2.

The Laplace spectrum ∆(ΩC) = ∪k,m∈Z+\0 √ k2 + m2. Question (M. Kac’66) Does ∆(Ω) determine Ω up to isometry? Weyl law (H. Weyl’11) N(λ) := # eigenvalues (w multiplicity) in (0, λ2], then lim

λ→∞ λ−1N(λ) = (4π)−1Area (Ω).

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 4 / 28

Can’t hear the shape of a drum!

Gordon–Webb-Wolpert’92

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 5 / 28

Can’t hear the shape of a drum!

Gordon–Webb-Wolpert’92

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 5 / 28

Can’t hear the shape of a drum!

Gordon–Webb-Wolpert’92 Consider domains with a smooth or an analytic boundary! Osgood-Phillips-Sarnak A C∞ isospectral set is compact. Conjecture (Sarnak’90) A C∞ isospectr. set consists of isolated points. Hezari-Zeldich, Popov-Topalov Analytic deformations of ellipses. Colin de Verdi` ere, Zeldich Generic analytic axis-symmetric domains.

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 6 / 28

Can’t hear the shape of a drum!

Gordon–Webb-Wolpert’92 Consider domains with a smooth or an analytic boundary! Osgood-Phillips-Sarnak A C∞ isospectral set is compact. Conjecture (Sarnak’90) A C∞ isospectr. set consists of isolated points. Hezari-Zeldich, Popov-Topalov Analytic deformations of ellipses. Colin de Verdi` ere, Zeldich Generic analytic axis-symmetric domains.

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 6 / 28

Can’t hear the shape of a drum!

Gordon–Webb-Wolpert’92 Consider domains with a smooth or an analytic boundary! Osgood-Phillips-Sarnak A C∞ isospectral set is compact. Conjecture (Sarnak’90) A C∞ isospectr. set consists of isolated points. Hezari-Zeldich, Popov-Topalov Analytic deformations of ellipses. Colin de Verdi` ere, Zeldich Generic analytic axis-symmetric domains.

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 6 / 28

Can’t hear the shape of a drum!

Gordon–Webb-Wolpert’92 Consider domains with a smooth or an analytic boundary! Osgood-Phillips-Sarnak A C∞ isospectral set is compact. Conjecture (Sarnak’90) A C∞ isospectr. set consists of isolated points. Hezari-Zeldich, Popov-Topalov Analytic deformations of ellipses. Colin de Verdi` ere, Zeldich Generic analytic axis-symmetric domains.

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 6 / 28

Can’t hear the shape of a drum!

Gordon–Webb-Wolpert’92 Consider domains with a smooth or an analytic boundary! Osgood-Phillips-Sarnak A C∞ isospectral set is compact. Conjecture (Sarnak’90) A C∞ isospectr. set consists of isolated points. Hezari-Zeldich, Popov-Topalov Analytic deformations of ellipses. Colin de Verdi` ere, Zeldich Generic analytic axis-symmetric domains.

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 6 / 28

Can’t hear the shape of a drum!

Gordon–Webb-Wolpert’92 Consider domains with a smooth or an analytic boundary! Osgood-Phillips-Sarnak A C∞ isospectral set is compact. Conjecture (Sarnak’90) A C∞ isospectr. set consists of isolated points. Hezari-Zeldich, Popov-Topalov Analytic deformations of ellipses. Colin de Verdi` ere, Zeldich Generic analytic axis-symmetric domains.

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 6 / 28

Can’t hear the shape of a drum!

Gordon–Webb-Wolpert’92 Consider domains with a smooth or an analytic boundary! Osgood-Phillips-Sarnak A C∞ isospectral set is compact. Conjecture (Sarnak’90) A C∞ isospectr. set consists of isolated points. Hezari-Zeldich, Popov-Topalov Analytic deformations of ellipses. Colin de Verdi` ere, Zeldich Generic analytic axis-symmetric domains.

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 6 / 28

Can you hear the shape of a Riemannian manifold?

Let (M, g) be a Riemannian compact manifold. Consider the spectrum

• f the Laplace-Beltrami operator ∆(M, g).

Question Does ∆(M, g) determine (M, g) up to an isometry? Sunada, Vingeras* ∃ isospectral sets of arbitrary finite cardinality. Conjecture (Sarnak’90) A C∞ isospectr. set consists of isolated points. Call Ω spectrally rigid (SR) if any smooth isopectral deformation {Ωt}t is an isometry, i.e. ∆(Ωt) ≡ ∆(Ω0). Conjecture (Sarnak’90) Any planar domain is SR.

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 7 / 28

Can you hear the shape of a Riemannian manifold?

Let (M, g) be a Riemannian compact manifold. Consider the spectrum

• f the Laplace-Beltrami operator ∆(M, g).

Question Does ∆(M, g) determine (M, g) up to an isometry? Sunada, Vingeras* ∃ isospectral sets of arbitrary finite cardinality. Conjecture (Sarnak’90) A C∞ isospectr. set consists of isolated points. Call Ω spectrally rigid (SR) if any smooth isopectral deformation {Ωt}t is an isometry, i.e. ∆(Ωt) ≡ ∆(Ω0). Conjecture (Sarnak’90) Any planar domain is SR.

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 7 / 28

Can you hear the shape of a Riemannian manifold?

Let (M, g) be a Riemannian compact manifold. Consider the spectrum

• f the Laplace-Beltrami operator ∆(M, g).

Question Does ∆(M, g) determine (M, g) up to an isometry? Sunada, Vingeras* ∃ isospectral sets of arbitrary finite cardinality. Conjecture (Sarnak’90) A C∞ isospectr. set consists of isolated points. Call Ω spectrally rigid (SR) if any smooth isopectral deformation {Ωt}t is an isometry, i.e. ∆(Ωt) ≡ ∆(Ω0). Conjecture (Sarnak’90) Any planar domain is SR.

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 7 / 28

Can you hear the shape of a Riemannian manifold?

Let (M, g) be a Riemannian compact manifold. Consider the spectrum

• f the Laplace-Beltrami operator ∆(M, g).

Question Does ∆(M, g) determine (M, g) up to an isometry? Sunada, Vingeras* ∃ isospectral sets of arbitrary finite cardinality. Conjecture (Sarnak’90) A C∞ isospectr. set consists of isolated points. Call Ω spectrally rigid (SR) if any smooth isopectral deformation {Ωt}t is an isometry, i.e. ∆(Ωt) ≡ ∆(Ω0). Conjecture (Sarnak’90) Any planar domain is SR.

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 7 / 28

Can you hear the shape of a Riemannian manifold?

Let (M, g) be a Riemannian compact manifold. Consider the spectrum

• f the Laplace-Beltrami operator ∆(M, g).

Question Does ∆(M, g) determine (M, g) up to an isometry? Sunada, Vingeras* ∃ isospectral sets of arbitrary finite cardinality. Conjecture (Sarnak’90) A C∞ isospectr. set consists of isolated points. Call Ω spectrally rigid (SR) if any smooth isopectral deformation {Ωt}t is an isometry, i.e. ∆(Ωt) ≡ ∆(Ω0). Conjecture (Sarnak’90) Any planar domain is SR.

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 7 / 28

Can you hear the shape of a Riemannian manifold?

Let (M, g) be a Riemannian compact manifold. Consider the spectrum

• f the Laplace-Beltrami operator ∆(M, g).

Question Does ∆(M, g) determine (M, g) up to an isometry? Sunada, Vingeras* ∃ isospectral sets of arbitrary finite cardinality. Conjecture (Sarnak’90) A C∞ isospectr. set consists of isolated points. Call Ω spectrally rigid (SR) if any smooth isopectral deformation {Ωt}t is an isometry, i.e. ∆(Ωt) ≡ ∆(Ω0). Conjecture (Sarnak’90) Any planar domain is SR.

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 7 / 28

Can you hear the shape of a Riemannian manifold?

Let (M, g) be a Riemannian compact manifold. Consider the spectrum

• f the Laplace-Beltrami operator ∆(M, g).

Question Does ∆(M, g) determine (M, g) up to an isometry? Sunada, Vingeras* ∃ isospectral sets of arbitrary finite cardinality. Conjecture (Sarnak’90) A C∞ isospectr. set consists of isolated points. Call Ω spectrally rigid (SR) if any smooth isopectral deformation {Ωt}t is an isometry, i.e. ∆(Ωt) ≡ ∆(Ω0). Conjecture (Sarnak’90) Any planar domain is SR.

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 7 / 28

Can you hear the shape of a Riemannian manifold?

Let (M, g) be a Riemannian compact manifold. Consider the spectrum

• f the Laplace-Beltrami operator ∆(M, g).

Question Does ∆(M, g) determine (M, g) up to an isometry? Sunada, Vingeras* ∃ isospectral sets of arbitrary finite cardinality. Conjecture (Sarnak’90) A C∞ isospectr. set consists of isolated points. Call Ω spectrally rigid (SR) if any smooth isopectral deformation {Ωt}t is an isometry, i.e. ∆(Ωt) ≡ ∆(Ω0). Conjecture (Sarnak’90) Any planar domain is SR.

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 7 / 28

Length spectrum

Let Ω ⊂ R2 be a strictly convex domain. Define the length spectrum L(Ω) := ∪PL(P) ∪ N L(∂Ω), L(P) – perimeter of a periodic orbit, ∪ – over all per orbits.

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 8 / 28

Length spectrum

Let Ω ⊂ R2 be a strictly convex domain. Define the length spectrum L(Ω) := ∪PL(P) ∪ N L(∂Ω), L(P) – perimeter of a periodic orbit, ∪ – over all per orbits.

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 8 / 28

Length spectrum

Let Ω ⊂ R2 be a strictly convex domain. Define the length spectrum L(Ω) := ∪PL(P) ∪ N L(∂Ω), L(P) – perimeter of a periodic orbit, ∪ – over all per orbits.

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 8 / 28

Length spectrum

Let Ω ⊂ R2 be a strictly convex domain. Define the length spectrum L(Ω) := ∪PL(P) ∪ N L(∂Ω), L(P) – perimeter of a periodic orbit, ∪ – over all per orbits.

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 8 / 28

Length spectrum

Let Ω ⊂ R2 be a strictly convex domain. Define the length spectrum L(Ω) := ∪PL(P) ∪ N L(∂Ω), L(P) – perimeter of a periodic orbit, ∪ – over all per orbits.

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 8 / 28

Length spectrum

Let Ω ⊂ R2 be a strictly convex domain. Define the length spectrum L(Ω) := ∪PL(P) ∪ N L(∂Ω), L(P) – perimeter of a periodic orbit, ∪ – over all per orbits.

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 9 / 28

Length spectrum

Let Ω ⊂ R2 be a strictly convex domain. Define the length spectrum L(Ω) := ∪PL(P) ∪ N L(∂Ω), L(P) – perimeter of a periodic orbit, ∪ – over all per orbits.

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 9 / 28

Length spectrum

Let Ω ⊂ R2 be a strictly convex domain. Define the length spectrum L(Ω) := ∪PL(P) ∪ N L(∂Ω), L(P) – perimeter of a periodic orbit, ∪ – over all per orbits.

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 9 / 28

Length spectrum

Let Ω ⊂ R2 be a strictly convex domain. Define the length spectrum L(Ω) := ∪PL(P) ∪ N L(∂Ω), L(P) – perimeter of a periodic orbit, ∪ – over all per orbits.

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 10 / 28

Length spectrum

Let Ω ⊂ R2 be a strictly convex domain. Define the length spectrum L(Ω) := ∪PL(P) ∪ N L(∂Ω), L(P) – perimeter of a periodic orbit, ∪ – over all per orbits.

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 11 / 28

Length spectrum

Let Ω ⊂ R2 be a strictly convex domain. Define the length spectrum L(Ω) := ∪PL(P) ∪ N L(∂Ω), L(P) – perimeter of a periodic orbit, ∪ – over all per orbits.

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 12 / 28

Length spectrum and Laplace spectrum

Let Ω ⊂ R2 be a strictly convex domain. Define the length spectrum L(Ω) := ∪PL(P) ∪ N L(∂Ω), L(P) – perimeter of a periodic orbit, ∪ – over all per orbits. Theorem (Chazarian, Anderson-Melrose, Guillemin, Duister- maat, ...) The Laplace ∆(Ω) determines the length L(Ω),

• generically. More exactly, the wave trace

w(t) = Re

• λj∈∆(Ω)

exp(iλjt) is C∞ outside of ±L(Ω) ∪ 0. Generically,

• sing. supp. of w(t) = ±L(Ω) ∪ 0.
• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 13 / 28

Length spectrum and Laplace spectrum

Let Ω ⊂ R2 be a strictly convex domain. Define the length spectrum L(Ω) := ∪PL(P) ∪ N L(∂Ω), L(P) – perimeter of a periodic orbit, ∪ – over all per orbits. Theorem (Chazarian, Anderson-Melrose, Guillemin, Duister- maat, ...) The Laplace ∆(Ω) determines the length L(Ω),

• generically. More exactly, the wave trace

w(t) = Re

• λj∈∆(Ω)

exp(iλjt) is C∞ outside of ±L(Ω) ∪ 0. Generically,

• sing. supp. of w(t) = ±L(Ω) ∪ 0.
• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 13 / 28

Length spectrum and Laplace spectrum

Let Ω ⊂ R2 be a strictly convex domain. Define the length spectrum L(Ω) := ∪PL(P) ∪ N L(∂Ω), L(P) – perimeter of a periodic orbit, ∪ – over all per orbits. Theorem (Chazarian, Anderson-Melrose, Guillemin, Duister- maat, ...) The Laplace ∆(Ω) determines the length L(Ω),

• generically. More exactly, the wave trace

w(t) = Re

• λj∈∆(Ω)

exp(iλjt) is C∞ outside of ±L(Ω) ∪ 0. Generically,

• sing. supp. of w(t) = ±L(Ω) ∪ 0.
• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 13 / 28

Length spectrum and Laplace spectrum

Let Ω ⊂ R2 be a strictly convex domain. Define the length spectrum L(Ω) := ∪PL(P) ∪ N L(∂Ω), L(P) – perimeter of a periodic orbit, ∪ – over all per orbits. Theorem (Chazarian, Anderson-Melrose, Guillemin, Duister- maat, ...) The Laplace ∆(Ω) determines the length L(Ω),

• generically. More exactly, the wave trace

w(t) = Re

• λj∈∆(Ω)

exp(iλjt) is C∞ outside of ±L(Ω) ∪ 0. Generically,

• sing. supp. of w(t) = ±L(Ω) ∪ 0.
• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 13 / 28

Length spectrum and Laplace spectrum

Let Ω ⊂ R2 be a strictly convex domain. Define the length spectrum L(Ω) := ∪PL(P) ∪ N L(∂Ω), L(P) – perimeter of a periodic orbit, ∪ – over all per orbits. Theorem (Chazarian, Anderson-Melrose, Guillemin, Duister- maat, ...) The Laplace ∆(Ω) determines the length L(Ω),

• generically. More exactly, the wave trace

w(t) = Re

• λj∈∆(Ω)

exp(iλjt) is C∞ outside of ±L(Ω) ∪ 0. Generically,

• sing. supp. of w(t) = ±L(Ω) ∪ 0.
• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 13 / 28

Length spectrum and Laplace spectrum

Let Ω ⊂ R2 be a strictly convex domain. Define the length spectrum L(Ω) := ∪PL(P) ∪ N L(∂Ω), L(P) – perimeter of a periodic orbit, ∪ – over all per orbits. Theorem (Chazarian, Anderson-Melrose, Guillemin, Duister- maat, ...) The Laplace ∆(Ω) determines the length L(Ω),

• generically. More exactly, the wave trace

w(t) = Re

• λj∈∆(Ω)

exp(iλjt) is C∞ outside of ±L(Ω) ∪ 0. Generically,

• sing. supp. of w(t) = ±L(Ω) ∪ 0.
• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 13 / 28

Length spectrum and Laplace spectrum

Let Ω ⊂ R2 be a strictly convex domain. Define the length spectrum L(Ω) := ∪PL(P) ∪ N L(∂Ω), L(P) – perimeter of a periodic orbit, ∪ – over all per orbits. Theorem (Chazarian, Anderson-Melrose, Guillemin, Duister- maat, ...) The Laplace ∆(Ω) determines the length L(Ω),

• generically. More exactly, the wave trace

w(t) = Re

• λj∈∆(Ω)

exp(iλjt) is C∞ outside of ±L(Ω) ∪ 0. Generically,

• sing. supp. of w(t) = ±L(Ω) ∪ 0.
• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 13 / 28

Can hear an axis-symmetric drum!

Call Ω dynamically spectrally rigid (DSR) if any smooth isopectral deformation {Ωt}t ⊂ Sr is an isometry, i.e. L(Ωt) ≡ L(Ω0). Theorem 1 De Simoi-K-Wei’16 A axis-symmetric domain near the circle is DSR. Theorem 2 (in progress) Callis-K-Sorrentino’19 A Cr generic axis-symmetric domain is DSR. More exactly, there is a Cr open and dense set of DSR axis-symmetric domains.

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 14 / 28

Can hear an axis-symmetric drum!

Call Ω dynamically spectrally rigid (DSR) if any smooth isopectral deformation {Ωt}t ⊂ Sr is an isometry, i.e. L(Ωt) ≡ L(Ω0). Theorem 1 De Simoi-K-Wei’16 A axis-symmetric domain near the circle is DSR. Theorem 2 (in progress) Callis-K-Sorrentino’19 A Cr generic axis-symmetric domain is DSR. More exactly, there is a Cr open and dense set of DSR axis-symmetric domains.

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 14 / 28

Can hear an axis-symmetric drum!

Call Ω dynamically spectrally rigid (DSR) if any smooth isopectral deformation {Ωt}t ⊂ Sr is an isometry, i.e. L(Ωt) ≡ L(Ω0). Theorem 1 De Simoi-K-Wei’16 A axis-symmetric domain near the circle is DSR. Theorem 2 (in progress) Callis-K-Sorrentino’19 A Cr generic axis-symmetric domain is DSR. More exactly, there is a Cr open and dense set of DSR axis-symmetric domains.

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 14 / 28

Can hear an axis-symmetric drum!

Call Ω dynamically spectrally rigid (DSR) if any smooth isopectral deformation {Ωt}t ⊂ Sr is an isometry, i.e. L(Ωt) ≡ L(Ω0). Theorem 1 De Simoi-K-Wei’16 A axis-symmetric domain near the circle is DSR. Theorem 2 (in progress) Callis-K-Sorrentino’19 A Cr generic axis-symmetric domain is DSR. More exactly, there is a Cr open and dense set of DSR axis-symmetric domains.

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 14 / 28

Can hear an axis-symmetric drum!

Call Ω dynamically spectrally rigid (DSR) if any smooth isopectral deformation {Ωt}t ⊂ Sr is an isometry, i.e. L(Ωt) ≡ L(Ω0). Theorem 1 De Simoi-K-Wei’16 A axis-symmetric domain near the circle is DSR. Theorem 2 (in progress) Callis-K-Sorrentino’19 A Cr generic axis-symmetric domain is DSR. More exactly, there is a Cr open and dense set of DSR axis-symmetric domains.

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 14 / 28

Can hear an axis-symmetric drum!

Call Ω dynamically spectrally rigid (DSR) if any smooth isopectral deformation {Ωt}t ⊂ Sr is an isometry, i.e. L(Ωt) ≡ L(Ω0). Theorem 1 De Simoi-K-Wei’16 A axis-symmetric domain near the circle is DSR. Theorem 2 (in progress) Callis-K-Sorrentino’19 A Cr generic axis-symmetric domain is DSR. More exactly, there is a Cr open and dense set of DSR axis-symmetric domains.

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 14 / 28

Can hear an axis-symmetric drum!

Call Ω dynamically spectrally rigid (DSR) if any smooth isopectral deformation {Ωt}t ⊂ Sr is an isometry, i.e. L(Ωt) ≡ L(Ω0). Theorem 1 De Simoi-K-Wei’16 A axis-symmetric domain near the circle is DSR. Theorem 2 (in progress) Callis-K-Sorrentino’19 A Cr generic axis-symmetric domain is DSR. More exactly, there is a Cr open and dense set of DSR axis-symmetric domains.

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 14 / 28

Can hear an axis-symmetric drum!

Call Ω dynamically spectrally rigid (DSR) if any smooth isopectral deformation {Ωt}t ⊂ Sr is an isometry, i.e. L(Ωt) ≡ L(Ω0). Theorem 1 De Simoi-K-Wei’16 A axis-symmetric domain near the circle is DSR. Theorem 2 (in progress) Callis-K-Sorrentino’19 A Cr generic axis-symmetric domain is DSR. More exactly, there is a Cr open and dense set of DSR axis-symmetric domains.

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 14 / 28

Can’t deform isospectrally a peicewise analytic Bunimovich drum!

Theorem 1 J. Chen-K-H. Zhang’ 19 A p.a. Bunimovich stadium is

• DSR. In addition, a p.a. Bunimovich squash-like stadium is DSR.

Similar to Hezari-Zeldich.

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 15 / 28

Can’t deform isospectrally a peicewise analytic Bunimovich drum!

Theorem 1 J. Chen-K-H. Zhang’ 19 A p.a. Bunimovich stadium is

• DSR. In addition, a p.a. Bunimovich squash-like stadium is DSR.

Similar to Hezari-Zeldich.

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 15 / 28

Can’t deform isospectrally a peicewise analytic Bunimovich drum!

Theorem 1 J. Chen-K-H. Zhang’ 19 A p.a. Bunimovich stadium is

• DSR. In addition, a p.a. Bunimovich squash-like stadium is DSR.

Similar to Hezari-Zeldich.

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 15 / 28

Can’t deform isospectrally a peicewise analytic Bunimovich drum!

Theorem 1 J. Chen-K-H. Zhang’ 19 A p.a. Bunimovich stadium is

• DSR. In addition, a p.a. Bunimovich squash-like stadium is DSR.

Similar to Hezari-Zeldich.

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 15 / 28

Can’t deform isospectrally a peicewise analytic Bunimovich drum!

Theorem 1 J. Chen-K-H. Zhang’ 19 A p.a. Bunimovich stadium is

• DSR. In addition, a p.a. Bunimovich squash-like stadium is DSR.

Similar to Hezari-Zeldich.

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 15 / 28

Three disks hyperbolic billiard

Balint=De Simoi-K-Leguil Marked Length Spectrum determins an analytic three disk system with Z2 × Z2 symmetries.

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 16 / 28

Three disks hyperbolic billiard

Balint=De Simoi-K-Leguil Marked Length Spectrum determins an analytic three disk system with Z2 × Z2 symmetries.

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 16 / 28

Three disks hyperbolic billiard

Balint=De Simoi-K-Leguil Marked Length Spectrum determins an analytic three disk system with Z2 × Z2 symmetries.

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 16 / 28

Marked Length spectrum

Let (S, g) be a negatively curved compact surface. Call the union of minimal geodesics in each homotopy class γ L(S, g) = ∪(ℓγ, γ) the marked length spectrum. Guillemin-Kazhdan’80 any (S, g) is spectrally rigid. Croke, Otal’90 the marked length spectrum determines (S, g) upto isometry. Croke-Sharafutdinov’98 the marked length spectrum determines negatively curved manifold (M, g) upto isometry.

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 17 / 28

Marked Length spectrum

Let (S, g) be a negatively curved compact surface. Call the union of minimal geodesics in each homotopy class γ L(S, g) = ∪(ℓγ, γ) the marked length spectrum. Guillemin-Kazhdan’80 any (S, g) is spectrally rigid. Croke, Otal’90 the marked length spectrum determines (S, g) upto isometry. Croke-Sharafutdinov’98 the marked length spectrum determines negatively curved manifold (M, g) upto isometry.

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 17 / 28

Marked Length spectrum

Let (S, g) be a negatively curved compact surface. Call the union of minimal geodesics in each homotopy class γ L(S, g) = ∪(ℓγ, γ) the marked length spectrum. Guillemin-Kazhdan’80 any (S, g) is spectrally rigid. Croke, Otal’90 the marked length spectrum determines (S, g) upto isometry. Croke-Sharafutdinov’98 the marked length spectrum determines negatively curved manifold (M, g) upto isometry.

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 17 / 28

Marked Length spectrum

Let (S, g) be a negatively curved compact surface. Call the union of minimal geodesics in each homotopy class γ L(S, g) = ∪(ℓγ, γ) the marked length spectrum. Guillemin-Kazhdan’80 any (S, g) is spectrally rigid. Croke, Otal’90 the marked length spectrum determines (S, g) upto isometry. Croke-Sharafutdinov’98 the marked length spectrum determines negatively curved manifold (M, g) upto isometry.

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 17 / 28

Marked Length spectrum

Let (S, g) be a negatively curved compact surface. Call the union of minimal geodesics in each homotopy class γ L(S, g) = ∪(ℓγ, γ) the marked length spectrum. Guillemin-Kazhdan’80 any (S, g) is spectrally rigid. Croke, Otal’90 the marked length spectrum determines (S, g) upto isometry. Croke-Sharafutdinov’98 the marked length spectrum determines negatively curved manifold (M, g) upto isometry.

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 17 / 28

Ideas of proof of Dynamical Spectral Rigidity

‘Skeleton’ of the dynamics. Birkhoff proved

Lemma

For any convex domain Ω and any q > 1 there is a periodic orbit of period q, given by inscribed q-gons and denoted Sq = Sq(Ω). If Ω is axis-symmetric, then Sq can be chosen axis-symmetric.

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 18 / 28

Ideas of proof of Dynamical Spectral Rigidity

‘Skeleton’ of the dynamics. Birkhoff proved

Lemma

For any convex domain Ω and any q > 1 there is a periodic orbit of period q, given by inscribed q-gons and denoted Sq = Sq(Ω). If Ω is axis-symmetric, then Sq can be chosen axis-symmetric.

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 18 / 28

Ideas of proof of Dynamical Spectral Rigidity

‘Skeleton’ of the dynamics. Birkhoff proved

Lemma

For any convex domain Ω and any q > 1 there is a periodic orbit of period q, given by inscribed q-gons and denoted Sq = Sq(Ω). If Ω is axis-symmetric, then Sq can be chosen axis-symmetric.

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 18 / 28

Ideas of proof of Dynamical Spectral Rigidity

‘Skeleton’ of the dynamics. Birkhoff proved

Lemma

For any convex domain Ω and any q > 1 there is a periodic orbit of period q, given by inscribed q-gons and denoted Sq = Sq(Ω). If Ω is axis-symmetric, then Sq can be chosen axis-symmetric.

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 18 / 28

Ideas of proof of Dynamical Spectral Rigidity

‘Skeleton’ of the dynamics. Birkhoff proved

Lemma

For any convex domain Ω and any q > 1 there is a periodic orbit of period q, given by inscribed q-gons and denoted Sq = Sq(Ω). If Ω is axis-symmetric, then Sq can be chosen axis-symmetric.

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 19 / 28

Ideas of proof of Dynamical Spectral Rigidity

‘Skeleton’ of the dynamics. Birkhoff proved

Lemma

For any convex domain Ω and any q > 1 there is a periodic orbit of period q, given by inscribed q-gons and denoted Sq = Sq(Ω). If Ω is axis-symmetric, then Sq can be chosen axis-symmetric.

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 20 / 28

Ideas of proof of Dynamical Spectral Rigidity

‘Skeleton’ of the dynamics: symmetric q-gons Sq = (x(k)

q , ϕ(k) q ), q > 1.Sr(T) – space of Cr-symmetric functions.

Consider an isospectral deformation {Ωt}t ⊂ Sr,

∂Ωt = ∂Ω0 + tn(s) + O(t2), n ∈ Sr(T).

Then ℓq(n) = q

k=1 n(x(k) q ) sin ϕ(k) q

= 0.

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 21 / 28

Ideas of proof of Dynamical Spectral Rigidity

‘Skeleton’ of the dynamics: symmetric q-gons Sq = (x(k)

q , ϕ(k) q ), q > 1.Sr(T) – space of Cr-symmetric functions.

Consider an isospectral deformation {Ωt}t ⊂ Sr,

∂Ωt = ∂Ω0 + tn(s) + O(t2), n ∈ Sr(T).

Then ℓq(n) = q

k=1 n(x(k) q ) sin ϕ(k) q

= 0.

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 21 / 28

Ideas of proof of Dynamical Spectral Rigidity

‘Skeleton’ of the dynamics: symmetric q-gons Sq = (x(k)

q , ϕ(k) q ), q > 1.Sr(T) – space of Cr-symmetric functions.

Consider an isospectral deformation {Ωt}t ⊂ Sr,

∂Ωt = ∂Ω0 + tn(s) + O(t2), n ∈ Sr(T).

Then ℓq(n) = q

k=1 n(x(k) q ) sin ϕ(k) q

= 0.

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 21 / 28

Ideas of proof of Dynamical Spectral Rigidity

‘Skeleton’ of the dynamics: symmetric q-gons Sq = (x(k)

q , ϕ(k) q ), q > 1.Sr(T) – space of Cr-symmetric functions.

Consider an isospectral deformation {Ωt}t ⊂ Sr,

∂Ωt = ∂Ω0 + tn(s) + O(t2), n ∈ Sr(T).

Then ℓq(n) = q

k=1 n(x(k) q ) sin ϕ(k) q

= 0.

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 21 / 28

Ideas of proof of Dynamical Spectral Rigidity

‘Skeleton’ of the dynamics: symmetric q-gons Sq = (x(k)

q , ϕ(k) q ), q > 1. Sr(T) – space of Cr-symmetric functions.

Consider an isospectral deformation {Ωt}t ⊂ Sr,

∂Ωt = ∂Ω0 + tn(s) + O(t2), n ∈ Sr(T).

Then ℓq(n) = q

k=1 n(x(k) q ) sin ϕ(k) q

= 0. Define a linearized isospectral operator

LΩ : Cr(T) → ℓ∞, LΩ(n) = (ℓq(n), q = 0, 1, . . . ).

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 22 / 28

Ideas of proof of Dynamical Spectral Rigidity

‘Skeleton’ of the dynamics: symmetric q-gons Sq = (x(k)

q , ϕ(k) q ), q > 1. Sr(T) – space of Cr-symmetric functions.

Consider an isospectral deformation {Ωt}t ⊂ Sr,

∂Ωt = ∂Ω0 + tn(s) + O(t2), n ∈ Sr(T).

Then ℓq(n) = q

k=1 n(x(k) q ) sin ϕ(k) q

= 0. Define a linearized isospectral operator

LΩ : Cr(T) → ℓ∞, LΩ(n) = (ℓq(n), q = 0, 1, . . . ).

Lemma

If LΩ is injective, then Ω is DSR.

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 23 / 28

Linearized Isospectral Operator for the circle

Consider an isospectral deformation {Ωt}t ⊂ Sr, of the circle. In polar coordinates (r, s) ∈ R+ × T ∂Ωt = {r = 1 + tn(s) + O(t2)}, n ∈ Sr(T). Then ℓq(n) =

q

• k=1

n(k q ) = 0.

Lemma

Let n(s) =

k∈Z+ nk cos ks be the Fourier expansion. Then ℓq(n) = 0

implies nkq = 0 for k ≥ 1.

Lemma

The Linearized Isospectral Operator LΩ0 is upper triangular with units

• n the diagonal.
• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 24 / 28

Linearized Isospectral Operator for the circle

Consider an isospectral deformation {Ωt}t ⊂ Sr, of the circle. In polar coordinates (r, s) ∈ R+ × T ∂Ωt = {r = 1 + tn(s) + O(t2)}, n ∈ Sr(T). Then ℓq(n) =

q

• k=1

n(k q ) = 0.

Lemma

Let n(s) =

k∈Z+ nk cos ks be the Fourier expansion. Then ℓq(n) = 0

implies nkq = 0 for k ≥ 1.

Lemma

The Linearized Isospectral Operator LΩ0 is upper triangular with units

• n the diagonal.
• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 24 / 28

Linearized Isospectral Operator for the circle

Consider an isospectral deformation {Ωt}t ⊂ Sr, of the circle. In polar coordinates (r, s) ∈ R+ × T ∂Ωt = {r = 1 + tn(s) + O(t2)}, n ∈ Sr(T). Then ℓq(n) =

q

• k=1

n(k q ) = 0.

Lemma

Let n(s) =

k∈Z+ nk cos ks be the Fourier expansion. Then ℓq(n) = 0

implies nkq = 0 for k ≥ 1.

Lemma

The Linearized Isospectral Operator LΩ0 is upper triangular with units

• n the diagonal.
• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 24 / 28

Linearized Isospectral Operator for the circle

Consider an isospectral deformation {Ωt}t ⊂ Sr, of the circle. In polar coordinates (r, s) ∈ R+ × T ∂Ωt = {r = 1 + tn(s) + O(t2)}, n ∈ Sr(T). Then ℓq(n) =

q

• k=1

n(k q ) = 0.

Lemma

Let n(s) =

k∈Z+ nk cos ks be the Fourier expansion. Then ℓq(n) = 0

implies nkq = 0 for k ≥ 1.

Lemma

The Linearized Isospectral Operator LΩ0 is upper triangular with units

• n the diagonal.
• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 24 / 28

Marvizi-Melroze invariants

Let Sq = (x(k)

q , ϕ(k) q ), q > 1 be symmetric maximal q-gons.

Pq be its perimeter. Marvizi-Melroze There are numbers {ck}k≥1 such that Pq ∼ c0 + c1 q2 + c2 q4 + c3 q6 + · · · , where for curvature ρ(s) we have c1 = −2

• ρ2/3(s)ds

c2 = 1 1080

• (9ρ4/3 + 8ρ−8/3 ˙

ρ2)(s)ds.

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 25 / 28

Marvizi-Melroze invariants

Let Sq = (x(k)

q , ϕ(k) q ), q > 1 be symmetric maximal q-gons.

Pq be its perimeter. Marvizi-Melroze There are numbers {ck}k≥1 such that Pq ∼ c0 + c1 q2 + c2 q4 + c3 q6 + · · · , where for curvature ρ(s) we have c1 = −2

• ρ2/3(s)ds

c2 = 1 1080

• (9ρ4/3 + 8ρ−8/3 ˙

ρ2)(s)ds.

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 25 / 28

Marvizi-Melroze invariants

Let Sq = (x(k)

q , ϕ(k) q ), q > 1 be symmetric maximal q-gons.

Pq be its perimeter. Marvizi-Melroze There are numbers {ck}k≥1 such that Pq ∼ c0 + c1 q2 + c2 q4 + c3 q6 + · · · , where for curvature ρ(s) we have c1 = −2

• ρ2/3(s)ds

c2 = 1 1080

• (9ρ4/3 + 8ρ−8/3 ˙

ρ2)(s)ds.

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 25 / 28

Marvizi-Melroze invariants

Let Sq = (x(k)

q , ϕ(k) q ), q > 1 be symmetric maximal q-gons.

Pq be its perimeter. Marvizi-Melroze There are numbers {ck}k≥1 such that Pq ∼ c0 + c1 q2 + c2 q4 + c3 q6 + · · · , where for curvature ρ(s) we have c1 = −2

• ρ2/3(s)ds

c2 = 1 1080

• (9ρ4/3 + 8ρ−8/3 ˙

ρ2)(s)ds.

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 25 / 28

Marvizi-Melroze invariants

Let Sq = (x(k)

q , ϕ(k) q ), q > 1 be symmetric maximal q-gons.

Pq be its perimeter. Marvizi-Melroze There are numbers {ck}k≥1 such that Pq ∼ c0 + c1 q2 + c2 q4 + c3 q6 + · · · , where for curvature ρ(s) we have c1 = −2

• ρ2/3(s)ds

c2 = 1 1080

• (9ρ4/3 + 8ρ−8/3 ˙

ρ2)(s)ds.

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 25 / 28

Marvizi-Melroze invariants

Let Sq = (x(k)

q , ϕ(k) q ), q > 1 be symmetric maximal q-gons.

Pq be its perimeter. Marvizi-Melroze There are numbers {ck}k≥1 such that Pq ∼ c0 + c1 q2 + c2 q4 + c3 q6 + · · · , where for curvature ρ(s) we have c1 = −2

• ρ2/3(s)ds

c2 = 1 1080

• (9ρ4/3 + 8ρ−8/3 ˙

ρ2)(s)ds.

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 25 / 28

Marvizi-Melroze invariants

Let Sq = (x(k)

q , ϕ(k) q ), q > 1 be symmetric maximal q-gons.

Pq be its perimeter. Marvizi-Melroze There are numbers {ck}k≥1 such that Pq ∼ c0 + c1 q2 + c2 q4 + c3 q6 + · · · , where for curvature ρ(s) we have c1 = −2

• ρ2/3(s)ds

c2 = 1 1080

• (9ρ4/3 + 8ρ−8/3 ˙

ρ2)(s)ds.

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 25 / 28

Marvizi-Melroze type invariants

(Sorrentino) Let L ∈ D be the Lazutkin coefficient associated to a caustic Γ(L). Then the length ℓ(Γ) = β0 + β1L1/3 + β2L2/3 + +β3L3/3 + · · · , where β0 = ℓ(∂Ω) (the perimeter of Ω) and βn =

• Θn(ρ, φ) dφ, where

Θn(ρ, φ) = Pn(ρ−1(φ)) · ρ(n)(φ) + Rn(ρ(φ)), where Pn is a non-zero polynomial in ρ−1.

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 26 / 28

Marvizi-Melroze type invariants

(Sorrentino) Let L ∈ D be the Lazutkin coefficient associated to a caustic Γ(L). Then the length ℓ(Γ) = β0 + β1L1/3 + β2L2/3 + +β3L3/3 + · · · , where β0 = ℓ(∂Ω) (the perimeter of Ω) and βn =

• Θn(ρ, φ) dφ, where

Θn(ρ, φ) = Pn(ρ−1(φ)) · ρ(n)(φ) + Rn(ρ(φ)), where Pn is a non-zero polynomial in ρ−1.

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 26 / 28

Marvizi-Melroze type invariants

(Sorrentino) Let L ∈ D be the Lazutkin coefficient associated to a caustic Γ(L). Then the length ℓ(Γ) = β0 + β1L1/3 + β2L2/3 + +β3L3/3 + · · · , where β0 = ℓ(∂Ω) (the perimeter of Ω) and βn =

• Θn(ρ, φ) dφ, where

Θn(ρ, φ) = Pn(ρ−1(φ)) · ρ(n)(φ) + Rn(ρ(φ)), where Pn is a non-zero polynomial in ρ−1.

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 26 / 28

Marvizi-Melroze type invariants

(Sorrentino) Let L ∈ D be the Lazutkin coefficient associated to a caustic Γ(L). Then the length ℓ(Γ) = β0 + β1L1/3 + β2L2/3 + +β3L3/3 + · · · , where β0 = ℓ(∂Ω) (the perimeter of Ω) and βn =

• Θn(ρ, φ) dφ, where

Θn(ρ, φ) = Pn(ρ−1(φ)) · ρ(n)(φ) + Rn(ρ(φ)), where Pn is a non-zero polynomial in ρ−1.

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 26 / 28

Marvizi-Melroze type invariants

(Sorrentino) Let L ∈ D be the Lazutkin coefficient associated to a caustic Γ(L). Then the length ℓ(Γ) = β0 + β1L1/3 + β2L2/3 + +β3L3/3 + · · · , where β0 = ℓ(∂Ω) (the perimeter of Ω) and βn =

• Θn(ρ, φ) dφ, where

Θn(ρ, φ) = Pn(ρ−1(φ)) · ρ(n)(φ) + Rn(ρ(φ)), where Pn is a non-zero polynomial in ρ−1.

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 26 / 28

Summary

• M. Kac’66 Can you hear the shape of a drum?

Gordon-Webb-Wolpert Counter-examples with peicewise smooth boundary. Conjecture (Sarnak’90) Any planar domain is spertrally rigidity De Simoi-K-Wei, Callis-K-Sorrentino Yes, for generic axis-symmetric domains

• J. Chen-K-H.K.Zhang Yes, for piecewice analytic

Bunimovich billiards

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 27 / 28

Summary

• M. Kac’66 Can you hear the shape of a drum?

Gordon-Webb-Wolpert Counter-examples with peicewise smooth boundary. Conjecture (Sarnak’90) Any planar domain is spertrally rigidity De Simoi-K-Wei, Callis-K-Sorrentino Yes, for generic axis-symmetric domains

• J. Chen-K-H.K.Zhang Yes, for piecewice analytic

Bunimovich billiards

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 27 / 28

Summary

• M. Kac’66 Can you hear the shape of a drum?

Gordon-Webb-Wolpert Counter-examples with peicewise smooth boundary. Conjecture (Sarnak’90) Any planar domain is spertrally rigidity De Simoi-K-Wei, Callis-K-Sorrentino Yes, for generic axis-symmetric domains

• J. Chen-K-H.K.Zhang Yes, for piecewice analytic

Bunimovich billiards

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 27 / 28

Summary

• M. Kac’66 Can you hear the shape of a drum?

Gordon-Webb-Wolpert Counter-examples with peicewise smooth boundary. Conjecture (Sarnak’90) Any planar domain is spertrally rigidity De Simoi-K-Wei, Callis-K-Sorrentino Yes, for generic axis-symmetric domains

• J. Chen-K-H.K.Zhang Yes, for piecewice analytic

Bunimovich billiards

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 27 / 28

Summary

• M. Kac’66 Can you hear the shape of a drum?

Gordon-Webb-Wolpert Counter-examples with peicewise smooth boundary. Conjecture (Sarnak’90) Any planar domain is spertrally rigidity De Simoi-K-Wei, Callis-K-Sorrentino Yes, for generic axis-symmetric domains

• J. Chen-K-H.K.Zhang Yes, for piecewice analytic

Bunimovich billiards

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 27 / 28

References

Jacopo De Simoi, Vadim Kaloshin, Qiaoling Wei, (with appedix co-authored with H. Hezari) Dynamical Spectral rigidity among Z2-symmetric strictly convex domains close to a circle, Annals of Math, 186–1, 2017, 1–40; Jianyu Chen, Vadim Kaloshin, Hong-Kun Zhang, Length spectrum rigidity for peicewide analytic Bunimovich billiards, preprint, 49pp. Jacopo De Simoi, Vadim Kaloshin, Martin Leguil, On the shape of chaotic billiards with analytic boundary , preprint, 50pp. Keagan Callis, Vadim Kaloshin, Alfonso Sorrentino, Marvizi-Melroze type invariants and dynamical spectral rigidity among generic Z2-symmetric convex domains, preprint, 32pp, in preparation.

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 28 / 28

References

Jacopo De Simoi, Vadim Kaloshin, Qiaoling Wei, (with appedix co-authored with H. Hezari) Dynamical Spectral rigidity among Z2-symmetric strictly convex domains close to a circle, Annals of Math, 186–1, 2017, 1–40; Jianyu Chen, Vadim Kaloshin, Hong-Kun Zhang, Length spectrum rigidity for peicewide analytic Bunimovich billiards, preprint, 49pp. Jacopo De Simoi, Vadim Kaloshin, Martin Leguil, On the shape of chaotic billiards with analytic boundary , preprint, 50pp. Keagan Callis, Vadim Kaloshin, Alfonso Sorrentino, Marvizi-Melroze type invariants and dynamical spectral rigidity among generic Z2-symmetric convex domains, preprint, 32pp, in preparation.

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 28 / 28

References

Jacopo De Simoi, Vadim Kaloshin, Qiaoling Wei, (with appedix co-authored with H. Hezari) Dynamical Spectral rigidity among Z2-symmetric strictly convex domains close to a circle, Annals of Math, 186–1, 2017, 1–40; Jianyu Chen, Vadim Kaloshin, Hong-Kun Zhang, Length spectrum rigidity for peicewide analytic Bunimovich billiards, preprint, 49pp. Jacopo De Simoi, Vadim Kaloshin, Martin Leguil, On the shape of chaotic billiards with analytic boundary , preprint, 50pp. Keagan Callis, Vadim Kaloshin, Alfonso Sorrentino, Marvizi-Melroze type invariants and dynamical spectral rigidity among generic Z2-symmetric convex domains, preprint, 32pp, in preparation.

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 28 / 28

References

Jacopo De Simoi, Vadim Kaloshin, Qiaoling Wei, (with appedix co-authored with H. Hezari) Dynamical Spectral rigidity among Z2-symmetric strictly convex domains close to a circle, Annals of Math, 186–1, 2017, 1–40; Jianyu Chen, Vadim Kaloshin, Hong-Kun Zhang, Length spectrum rigidity for peicewide analytic Bunimovich billiards, preprint, 49pp. Jacopo De Simoi, Vadim Kaloshin, Martin Leguil, On the shape of chaotic billiards with analytic boundary , preprint, 50pp. Keagan Callis, Vadim Kaloshin, Alfonso Sorrentino, Marvizi-Melroze type invariants and dynamical spectral rigidity among generic Z2-symmetric convex domains, preprint, 32pp, in preparation.

• V. Kaloshin (the ETH-ITS & U of Maryland)

Spectral Rigidity February 7, 2019 28 / 28