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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

  1. Can you hear the shape of a drum? Consider the Dirichlet problem in a domain Ω ⊂ R 2 . � ∆ u + λ 2 u = 0 u | ∂ Ω = 0 . ∆(Ω) := { 0 < λ 1 ≤ λ 2 ≤ · · · } — Laplace spectrum. Example 1 Let Ω C = [ 0 , π ] × [ 0 , π ] ∋ ( x , y ) . For any pair k , m ∈ Z + \ 0 let � k 2 + m 2 . u ( x , y ) = sin kx · sin my and λ = √ k 2 + m 2 . The Laplace spectrum ∆(Ω C ) = ∪ k , m ∈ Z + \ 0 Question (M. Kac’66) Does ∆(Ω) determine Ω up to isometry? Weyl law (H. Weyl’11) N ( λ ) := # eigenvalues (w multiplicity) in ( 0 , λ 2 ] , then λ →∞ λ − 1 N ( λ ) = ( 4 π ) − 1 Area (Ω) . lim V. Kaloshin (the ETH-ITS & U of Maryland) Spectral Rigidity February 7, 2019 4 / 28

  2. Can you hear the shape of a drum? Consider the Dirichlet problem in a domain Ω ⊂ R 2 . � ∆ u + λ 2 u = 0 u | ∂ Ω = 0 . ∆(Ω) := { 0 < λ 1 ≤ λ 2 ≤ · · · } — Laplace spectrum. Example 1 Let Ω C = [ 0 , π ] × [ 0 , π ] ∋ ( x , y ) . For any pair k , m ∈ Z + \ 0 let � k 2 + m 2 . u ( x , y ) = sin kx · sin my and λ = √ k 2 + m 2 . The Laplace spectrum ∆(Ω C ) = ∪ k , m ∈ Z + \ 0 Question (M. Kac’66) Does ∆(Ω) determine Ω up to isometry? Weyl law (H. Weyl’11) N ( λ ) := # eigenvalues (w multiplicity) in ( 0 , λ 2 ] , then λ →∞ λ − 1 N ( λ ) = ( 4 π ) − 1 Area (Ω) . lim V. Kaloshin (the ETH-ITS & U of Maryland) Spectral Rigidity February 7, 2019 4 / 28

  3. Can you hear the shape of a drum? Consider the Dirichlet problem in a domain Ω ⊂ R 2 . � ∆ u + λ 2 u = 0 u | ∂ Ω = 0 . ∆(Ω) := { 0 < λ 1 ≤ λ 2 ≤ · · · } — Laplace spectrum. Example 1 Let Ω C = [ 0 , π ] × [ 0 , π ] ∋ ( x , y ) . For any pair k , m ∈ Z + \ 0 let � k 2 + m 2 . u ( x , y ) = sin kx · sin my and λ = √ k 2 + m 2 . The Laplace spectrum ∆(Ω C ) = ∪ k , m ∈ Z + \ 0 Question (M. Kac’66) Does ∆(Ω) determine Ω up to isometry? Weyl law (H. Weyl’11) N ( λ ) := # eigenvalues (w multiplicity) in ( 0 , λ 2 ] , then λ →∞ λ − 1 N ( λ ) = ( 4 π ) − 1 Area (Ω) . lim V. Kaloshin (the ETH-ITS & U of Maryland) Spectral Rigidity February 7, 2019 4 / 28

  4. Can you hear the shape of a drum? Consider the Dirichlet problem in a domain Ω ⊂ R 2 . � ∆ u + λ 2 u = 0 u | ∂ Ω = 0 . ∆(Ω) := { 0 < λ 1 ≤ λ 2 ≤ · · · } — Laplace spectrum. Example 1 Let Ω C = [ 0 , π ] × [ 0 , π ] ∋ ( x , y ) . For any pair k , m ∈ Z + \ 0 let � k 2 + m 2 . u ( x , y ) = sin kx · sin my and λ = √ k 2 + m 2 . The Laplace spectrum ∆(Ω C ) = ∪ k , m ∈ Z + \ 0 Question (M. Kac’66) Does ∆(Ω) determine Ω up to isometry? Weyl law (H. Weyl’11) N ( λ ) := # eigenvalues (w multiplicity) in ( 0 , λ 2 ] , then λ →∞ λ − 1 N ( λ ) = ( 4 π ) − 1 Area (Ω) . lim V. Kaloshin (the ETH-ITS & U of Maryland) Spectral Rigidity February 7, 2019 4 / 28

  5. 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

  6. 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

  7. 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

  8. 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

  9. 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

  10. 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

  11. 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

  12. 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

  13. 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

  14. Can you hear the shape of a Riemannian manifold? Let ( M , g ) be a Riemannian compact manifold. Consider the spectrum of 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 { Ω t } t is an isometry, i.e. ∆(Ω t ) ≡ ∆(Ω 0 ) . Conjecture (Sarnak’90) Any planar domain is SR. V. Kaloshin (the ETH-ITS & U of Maryland) Spectral Rigidity February 7, 2019 7 / 28

  15. Can you hear the shape of a Riemannian manifold? Let ( M , g ) be a Riemannian compact manifold. Consider the spectrum of 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 { Ω t } t is an isometry, i.e. ∆(Ω t ) ≡ ∆(Ω 0 ) . Conjecture (Sarnak’90) Any planar domain is SR. V. Kaloshin (the ETH-ITS & U of Maryland) Spectral Rigidity February 7, 2019 7 / 28

  16. Can you hear the shape of a Riemannian manifold? Let ( M , g ) be a Riemannian compact manifold. Consider the spectrum of 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 { Ω t } t is an isometry, i.e. ∆(Ω t ) ≡ ∆(Ω 0 ) . Conjecture (Sarnak’90) Any planar domain is SR. V. Kaloshin (the ETH-ITS & U of Maryland) Spectral Rigidity February 7, 2019 7 / 28

  17. Can you hear the shape of a Riemannian manifold? Let ( M , g ) be a Riemannian compact manifold. Consider the spectrum of 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 { Ω t } t is an isometry, i.e. ∆(Ω t ) ≡ ∆(Ω 0 ) . Conjecture (Sarnak’90) Any planar domain is SR. V. Kaloshin (the ETH-ITS & U of Maryland) Spectral Rigidity February 7, 2019 7 / 28

  18. Can you hear the shape of a Riemannian manifold? Let ( M , g ) be a Riemannian compact manifold. Consider the spectrum of 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 { Ω t } t is an isometry, i.e. ∆(Ω t ) ≡ ∆(Ω 0 ) . Conjecture (Sarnak’90) Any planar domain is SR. V. Kaloshin (the ETH-ITS & U of Maryland) Spectral Rigidity February 7, 2019 7 / 28

  19. Can you hear the shape of a Riemannian manifold? Let ( M , g ) be a Riemannian compact manifold. Consider the spectrum of 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 { Ω t } t is an isometry, i.e. ∆(Ω t ) ≡ ∆(Ω 0 ) . Conjecture (Sarnak’90) Any planar domain is SR. V. Kaloshin (the ETH-ITS & U of Maryland) Spectral Rigidity February 7, 2019 7 / 28

  20. Can you hear the shape of a Riemannian manifold? Let ( M , g ) be a Riemannian compact manifold. Consider the spectrum of 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 { Ω t } t is an isometry, i.e. ∆(Ω t ) ≡ ∆(Ω 0 ) . Conjecture (Sarnak’90) Any planar domain is SR. V. Kaloshin (the ETH-ITS & U of Maryland) Spectral Rigidity February 7, 2019 7 / 28

  21. Can you hear the shape of a Riemannian manifold? Let ( M , g ) be a Riemannian compact manifold. Consider the spectrum of 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 { Ω t } t is an isometry, i.e. ∆(Ω t ) ≡ ∆(Ω 0 ) . Conjecture (Sarnak’90) Any planar domain is SR. V. Kaloshin (the ETH-ITS & U of Maryland) Spectral Rigidity February 7, 2019 7 / 28

  22. Length spectrum Let Ω ⊂ R 2 be a strictly convex domain. Define the length spectrum L (Ω) := ∪ P L ( 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

  23. Length spectrum Let Ω ⊂ R 2 be a strictly convex domain. Define the length spectrum L (Ω) := ∪ P L ( 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

  24. Length spectrum Let Ω ⊂ R 2 be a strictly convex domain. Define the length spectrum L (Ω) := ∪ P L ( 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

  25. Length spectrum Let Ω ⊂ R 2 be a strictly convex domain. Define the length spectrum L (Ω) := ∪ P L ( 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

  26. Length spectrum Let Ω ⊂ R 2 be a strictly convex domain. Define the length spectrum L (Ω) := ∪ P L ( 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

  27. Length spectrum Let Ω ⊂ R 2 be a strictly convex domain. Define the length spectrum L (Ω) := ∪ P L ( 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

  28. Length spectrum Let Ω ⊂ R 2 be a strictly convex domain. Define the length spectrum L (Ω) := ∪ P L ( 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

  29. Length spectrum Let Ω ⊂ R 2 be a strictly convex domain. Define the length spectrum L (Ω) := ∪ P L ( 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

  30. Length spectrum Let Ω ⊂ R 2 be a strictly convex domain. Define the length spectrum L (Ω) := ∪ P L ( 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

  31. Length spectrum Let Ω ⊂ R 2 be a strictly convex domain. Define the length spectrum L (Ω) := ∪ P L ( 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

  32. Length spectrum Let Ω ⊂ R 2 be a strictly convex domain. Define the length spectrum L (Ω) := ∪ P L ( 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

  33. Length spectrum and Laplace spectrum Let Ω ⊂ R 2 be a strictly convex domain. Define the length spectrum L (Ω) := ∪ P L ( 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 exp( i λ j t ) λ j ∈ ∆(Ω) 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

  34. Length spectrum and Laplace spectrum Let Ω ⊂ R 2 be a strictly convex domain. Define the length spectrum L (Ω) := ∪ P L ( 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 exp( i λ j t ) λ j ∈ ∆(Ω) 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

  35. Length spectrum and Laplace spectrum Let Ω ⊂ R 2 be a strictly convex domain. Define the length spectrum L (Ω) := ∪ P L ( 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 exp( i λ j t ) λ j ∈ ∆(Ω) 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

  36. Length spectrum and Laplace spectrum Let Ω ⊂ R 2 be a strictly convex domain. Define the length spectrum L (Ω) := ∪ P L ( 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 exp( i λ j t ) λ j ∈ ∆(Ω) 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

  37. Length spectrum and Laplace spectrum Let Ω ⊂ R 2 be a strictly convex domain. Define the length spectrum L (Ω) := ∪ P L ( 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 exp( i λ j t ) λ j ∈ ∆(Ω) 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

  38. Length spectrum and Laplace spectrum Let Ω ⊂ R 2 be a strictly convex domain. Define the length spectrum L (Ω) := ∪ P L ( 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 exp( i λ j t ) λ j ∈ ∆(Ω) 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

  39. Length spectrum and Laplace spectrum Let Ω ⊂ R 2 be a strictly convex domain. Define the length spectrum L (Ω) := ∪ P L ( 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 exp( i λ j t ) λ j ∈ ∆(Ω) 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

  40. Can hear an axis-symmetric drum! Call Ω dynamically spectrally rigid (DSR) if any smooth isopectral deformation { Ω t } t ⊂ S r is an isometry, i.e. L (Ω t ) ≡ L (Ω 0 ) . 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 C r generic axis-symmetric domain is DSR. More exactly, there is a C r open and dense set of DSR axis-symmetric domains. V. Kaloshin (the ETH-ITS & U of Maryland) Spectral Rigidity February 7, 2019 14 / 28

  41. Can hear an axis-symmetric drum! Call Ω dynamically spectrally rigid (DSR) if any smooth isopectral deformation { Ω t } t ⊂ S r is an isometry, i.e. L (Ω t ) ≡ L (Ω 0 ) . 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 C r generic axis-symmetric domain is DSR. More exactly, there is a C r open and dense set of DSR axis-symmetric domains. V. Kaloshin (the ETH-ITS & U of Maryland) Spectral Rigidity February 7, 2019 14 / 28

  42. Can hear an axis-symmetric drum! Call Ω dynamically spectrally rigid (DSR) if any smooth isopectral deformation { Ω t } t ⊂ S r is an isometry, i.e. L (Ω t ) ≡ L (Ω 0 ) . 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 C r generic axis-symmetric domain is DSR. More exactly, there is a C r open and dense set of DSR axis-symmetric domains. V. Kaloshin (the ETH-ITS & U of Maryland) Spectral Rigidity February 7, 2019 14 / 28

  43. Can hear an axis-symmetric drum! Call Ω dynamically spectrally rigid (DSR) if any smooth isopectral deformation { Ω t } t ⊂ S r is an isometry, i.e. L (Ω t ) ≡ L (Ω 0 ) . 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 C r generic axis-symmetric domain is DSR. More exactly, there is a C r open and dense set of DSR axis-symmetric domains. V. Kaloshin (the ETH-ITS & U of Maryland) Spectral Rigidity February 7, 2019 14 / 28

  44. Can hear an axis-symmetric drum! Call Ω dynamically spectrally rigid (DSR) if any smooth isopectral deformation { Ω t } t ⊂ S r is an isometry, i.e. L (Ω t ) ≡ L (Ω 0 ) . 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 C r generic axis-symmetric domain is DSR. More exactly, there is a C r open and dense set of DSR axis-symmetric domains. V. Kaloshin (the ETH-ITS & U of Maryland) Spectral Rigidity February 7, 2019 14 / 28

  45. Can hear an axis-symmetric drum! Call Ω dynamically spectrally rigid (DSR) if any smooth isopectral deformation { Ω t } t ⊂ S r is an isometry, i.e. L (Ω t ) ≡ L (Ω 0 ) . 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 C r generic axis-symmetric domain is DSR. More exactly, there is a C r open and dense set of DSR axis-symmetric domains. V. Kaloshin (the ETH-ITS & U of Maryland) Spectral Rigidity February 7, 2019 14 / 28

  46. Can hear an axis-symmetric drum! Call Ω dynamically spectrally rigid (DSR) if any smooth isopectral deformation { Ω t } t ⊂ S r is an isometry, i.e. L (Ω t ) ≡ L (Ω 0 ) . 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 C r generic axis-symmetric domain is DSR. More exactly, there is a C r open and dense set of DSR axis-symmetric domains. V. Kaloshin (the ETH-ITS & U of Maryland) Spectral Rigidity February 7, 2019 14 / 28

  47. Can hear an axis-symmetric drum! Call Ω dynamically spectrally rigid (DSR) if any smooth isopectral deformation { Ω t } t ⊂ S r is an isometry, i.e. L (Ω t ) ≡ L (Ω 0 ) . 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 C r generic axis-symmetric domain is DSR. More exactly, there is a C r open and dense set of DSR axis-symmetric domains. V. Kaloshin (the ETH-ITS & U of Maryland) Spectral Rigidity February 7, 2019 14 / 28

  48. 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

  49. 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

  50. 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

  51. 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

  52. 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

  53. Three disks hyperbolic billiard Balint=De Simoi-K-Leguil Marked Length Spectrum determins an analytic three disk system with Z 2 × Z 2 symmetries. V. Kaloshin (the ETH-ITS & U of Maryland) Spectral Rigidity February 7, 2019 16 / 28

  54. Three disks hyperbolic billiard Balint=De Simoi-K-Leguil Marked Length Spectrum determins an analytic three disk system with Z 2 × Z 2 symmetries. V. Kaloshin (the ETH-ITS & U of Maryland) Spectral Rigidity February 7, 2019 16 / 28

  55. Three disks hyperbolic billiard Balint=De Simoi-K-Leguil Marked Length Spectrum determins an analytic three disk system with Z 2 × Z 2 symmetries. V. Kaloshin (the ETH-ITS & U of Maryland) Spectral Rigidity February 7, 2019 16 / 28

  56. 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 ) = ∪ ( ℓ γ , 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

  57. 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 ) = ∪ ( ℓ γ , 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

  58. 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 ) = ∪ ( ℓ γ , 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

  59. 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 ) = ∪ ( ℓ γ , 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

  60. 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 ) = ∪ ( ℓ γ , 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

  61. 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 S q = S q (Ω) . If Ω is axis-symmetric, then S q can be chosen axis-symmetric. V. Kaloshin (the ETH-ITS & U of Maryland) Spectral Rigidity February 7, 2019 18 / 28

  62. 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 S q = S q (Ω) . If Ω is axis-symmetric, then S q can be chosen axis-symmetric. V. Kaloshin (the ETH-ITS & U of Maryland) Spectral Rigidity February 7, 2019 18 / 28

  63. 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 S q = S q (Ω) . If Ω is axis-symmetric, then S q can be chosen axis-symmetric. V. Kaloshin (the ETH-ITS & U of Maryland) Spectral Rigidity February 7, 2019 18 / 28

  64. 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 S q = S q (Ω) . If Ω is axis-symmetric, then S q can be chosen axis-symmetric. V. Kaloshin (the ETH-ITS & U of Maryland) Spectral Rigidity February 7, 2019 18 / 28

  65. 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 S q = S q (Ω) . If Ω is axis-symmetric, then S q can be chosen axis-symmetric. V. Kaloshin (the ETH-ITS & U of Maryland) Spectral Rigidity February 7, 2019 19 / 28

  66. 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 S q = S q (Ω) . If Ω is axis-symmetric, then S q can be chosen axis-symmetric. V. Kaloshin (the ETH-ITS & U of Maryland) Spectral Rigidity February 7, 2019 20 / 28

  67. Ideas of proof of Dynamical Spectral Rigidity ‘Skeleton’ of the dynamics: symmetric q -gons S q = ( x ( k ) q , ϕ ( k ) q ) , q > 1. S r ( T ) – space of C r -symmetric functions. Consider an isospectral deformation { Ω t } t ⊂ S r , ∂ Ω t = ∂ Ω 0 + tn ( s ) + O ( t 2 ) , n ∈ S r ( T ) . k = 1 n ( x ( k ) q ) sin ϕ ( k ) Then ℓ q ( n ) = � q = 0 . q V. Kaloshin (the ETH-ITS & U of Maryland) Spectral Rigidity February 7, 2019 21 / 28

  68. Ideas of proof of Dynamical Spectral Rigidity ‘Skeleton’ of the dynamics: symmetric q -gons S q = ( x ( k ) q , ϕ ( k ) q ) , q > 1. S r ( T ) – space of C r -symmetric functions. Consider an isospectral deformation { Ω t } t ⊂ S r , ∂ Ω t = ∂ Ω 0 + tn ( s ) + O ( t 2 ) , n ∈ S r ( T ) . k = 1 n ( x ( k ) q ) sin ϕ ( k ) Then ℓ q ( n ) = � q = 0 . q V. Kaloshin (the ETH-ITS & U of Maryland) Spectral Rigidity February 7, 2019 21 / 28

  69. Ideas of proof of Dynamical Spectral Rigidity ‘Skeleton’ of the dynamics: symmetric q -gons S q = ( x ( k ) q , ϕ ( k ) q ) , q > 1. S r ( T ) – space of C r -symmetric functions. Consider an isospectral deformation { Ω t } t ⊂ S r , ∂ Ω t = ∂ Ω 0 + tn ( s ) + O ( t 2 ) , n ∈ S r ( T ) . k = 1 n ( x ( k ) q ) sin ϕ ( k ) Then ℓ q ( n ) = � q = 0 . q V. Kaloshin (the ETH-ITS & U of Maryland) Spectral Rigidity February 7, 2019 21 / 28

  70. Ideas of proof of Dynamical Spectral Rigidity ‘Skeleton’ of the dynamics: symmetric q -gons S q = ( x ( k ) q , ϕ ( k ) q ) , q > 1. S r ( T ) – space of C r -symmetric functions. Consider an isospectral deformation { Ω t } t ⊂ S r , ∂ Ω t = ∂ Ω 0 + tn ( s ) + O ( t 2 ) , n ∈ S r ( T ) . k = 1 n ( x ( k ) q ) sin ϕ ( k ) Then ℓ q ( n ) = � q = 0 . q V. Kaloshin (the ETH-ITS & U of Maryland) Spectral Rigidity February 7, 2019 21 / 28

  71. Ideas of proof of Dynamical Spectral Rigidity ‘Skeleton’ of the dynamics: symmetric q -gons S q = ( x ( k ) q , ϕ ( k ) q ) , q > 1. S r ( T ) – space of C r -symmetric functions. Consider an isospectral deformation { Ω t } t ⊂ S r , ∂ Ω t = ∂ Ω 0 + tn ( s ) + O ( t 2 ) , n ∈ S r ( T ) . ℓ q ( n ) = � q k = 1 n ( x ( k ) q ) sin ϕ ( k ) Then = 0 . q Define a linearized isospectral operator L Ω : C r ( T ) → ℓ ∞ , L Ω ( n ) = ( ℓ q ( n ) , q = 0 , 1 , . . . ) . V. Kaloshin (the ETH-ITS & U of Maryland) Spectral Rigidity February 7, 2019 22 / 28

  72. Ideas of proof of Dynamical Spectral Rigidity ‘Skeleton’ of the dynamics: symmetric q -gons S q = ( x ( k ) q , ϕ ( k ) q ) , q > 1. S r ( T ) – space of C r -symmetric functions. Consider an isospectral deformation { Ω t } t ⊂ S r , ∂ Ω t = ∂ Ω 0 + tn ( s ) + O ( t 2 ) , n ∈ S r ( T ) . ℓ q ( n ) = � q k = 1 n ( x ( k ) q ) sin ϕ ( k ) Then = 0 . q Define a linearized isospectral operator L Ω : C r ( T ) → ℓ ∞ , L Ω ( n ) = ( ℓ q ( n ) , q = 0 , 1 , . . . ) . Lemma If L Ω is injective, then Ω is DSR. V. Kaloshin (the ETH-ITS & U of Maryland) Spectral Rigidity February 7, 2019 23 / 28

  73. Linearized Isospectral Operator for the circle Consider an isospectral deformation { Ω t } t ⊂ S r , of the circle. In polar coordinates ( r , s ) ∈ R + × T ∂ Ω t = { r = 1 + tn ( s ) + O ( t 2 ) } , n ∈ S r ( T ) . Then q n ( k � ℓ q ( n ) = q ) = 0 . k = 1 Lemma Let n ( s ) = � k ∈ Z + n k cos ks be the Fourier expansion. Then ℓ q ( n ) = 0 implies n kq = 0 for k ≥ 1 . Lemma The Linearized Isospectral Operator L Ω 0 is upper triangular with units on the diagonal. V. Kaloshin (the ETH-ITS & U of Maryland) Spectral Rigidity February 7, 2019 24 / 28

  74. Linearized Isospectral Operator for the circle Consider an isospectral deformation { Ω t } t ⊂ S r , of the circle. In polar coordinates ( r , s ) ∈ R + × T ∂ Ω t = { r = 1 + tn ( s ) + O ( t 2 ) } , n ∈ S r ( T ) . Then q n ( k � ℓ q ( n ) = q ) = 0 . k = 1 Lemma Let n ( s ) = � k ∈ Z + n k cos ks be the Fourier expansion. Then ℓ q ( n ) = 0 implies n kq = 0 for k ≥ 1 . Lemma The Linearized Isospectral Operator L Ω 0 is upper triangular with units on the diagonal. V. Kaloshin (the ETH-ITS & U of Maryland) Spectral Rigidity February 7, 2019 24 / 28

  75. Linearized Isospectral Operator for the circle Consider an isospectral deformation { Ω t } t ⊂ S r , of the circle. In polar coordinates ( r , s ) ∈ R + × T ∂ Ω t = { r = 1 + tn ( s ) + O ( t 2 ) } , n ∈ S r ( T ) . Then q n ( k � ℓ q ( n ) = q ) = 0 . k = 1 Lemma Let n ( s ) = � k ∈ Z + n k cos ks be the Fourier expansion. Then ℓ q ( n ) = 0 implies n kq = 0 for k ≥ 1 . Lemma The Linearized Isospectral Operator L Ω 0 is upper triangular with units on the diagonal. V. Kaloshin (the ETH-ITS & U of Maryland) Spectral Rigidity February 7, 2019 24 / 28

  76. Linearized Isospectral Operator for the circle Consider an isospectral deformation { Ω t } t ⊂ S r , of the circle. In polar coordinates ( r , s ) ∈ R + × T ∂ Ω t = { r = 1 + tn ( s ) + O ( t 2 ) } , n ∈ S r ( T ) . Then q n ( k � ℓ q ( n ) = q ) = 0 . k = 1 Lemma Let n ( s ) = � k ∈ Z + n k cos ks be the Fourier expansion. Then ℓ q ( n ) = 0 implies n kq = 0 for k ≥ 1 . Lemma The Linearized Isospectral Operator L Ω 0 is upper triangular with units on the diagonal. V. Kaloshin (the ETH-ITS & U of Maryland) Spectral Rigidity February 7, 2019 24 / 28

  77. Marvizi-Melroze invariants Let S q = ( x ( k ) q , ϕ ( k ) q ) , q > 1 be symmetric maximal q -gons. P q be its perimeter. Marvizi-Melroze There are numbers { c k } k ≥ 1 such that P q ∼ c 0 + c 1 q 2 + c 2 q 4 + c 3 q 6 + · · · , where for curvature ρ ( s ) we have � ρ 2 / 3 ( s ) ds c 1 = − 2 1 � ( 9 ρ 4 / 3 + 8 ρ − 8 / 3 ˙ ρ 2 )( s ) ds . c 2 = 1080 V. Kaloshin (the ETH-ITS & U of Maryland) Spectral Rigidity February 7, 2019 25 / 28

  78. Marvizi-Melroze invariants Let S q = ( x ( k ) q , ϕ ( k ) q ) , q > 1 be symmetric maximal q -gons. P q be its perimeter. Marvizi-Melroze There are numbers { c k } k ≥ 1 such that P q ∼ c 0 + c 1 q 2 + c 2 q 4 + c 3 q 6 + · · · , where for curvature ρ ( s ) we have � ρ 2 / 3 ( s ) ds c 1 = − 2 1 � ( 9 ρ 4 / 3 + 8 ρ − 8 / 3 ˙ ρ 2 )( s ) ds . c 2 = 1080 V. Kaloshin (the ETH-ITS & U of Maryland) Spectral Rigidity February 7, 2019 25 / 28

  79. Marvizi-Melroze invariants Let S q = ( x ( k ) q , ϕ ( k ) q ) , q > 1 be symmetric maximal q -gons. P q be its perimeter. Marvizi-Melroze There are numbers { c k } k ≥ 1 such that P q ∼ c 0 + c 1 q 2 + c 2 q 4 + c 3 q 6 + · · · , where for curvature ρ ( s ) we have � ρ 2 / 3 ( s ) ds c 1 = − 2 1 � ( 9 ρ 4 / 3 + 8 ρ − 8 / 3 ˙ ρ 2 )( s ) ds . c 2 = 1080 V. Kaloshin (the ETH-ITS & U of Maryland) Spectral Rigidity February 7, 2019 25 / 28

  80. Marvizi-Melroze invariants Let S q = ( x ( k ) q , ϕ ( k ) q ) , q > 1 be symmetric maximal q -gons. P q be its perimeter. Marvizi-Melroze There are numbers { c k } k ≥ 1 such that P q ∼ c 0 + c 1 q 2 + c 2 q 4 + c 3 q 6 + · · · , where for curvature ρ ( s ) we have � ρ 2 / 3 ( s ) ds c 1 = − 2 1 � ( 9 ρ 4 / 3 + 8 ρ − 8 / 3 ˙ ρ 2 )( s ) ds . c 2 = 1080 V. Kaloshin (the ETH-ITS & U of Maryland) Spectral Rigidity February 7, 2019 25 / 28

  81. Marvizi-Melroze invariants Let S q = ( x ( k ) q , ϕ ( k ) q ) , q > 1 be symmetric maximal q -gons. P q be its perimeter. Marvizi-Melroze There are numbers { c k } k ≥ 1 such that P q ∼ c 0 + c 1 q 2 + c 2 q 4 + c 3 q 6 + · · · , where for curvature ρ ( s ) we have � ρ 2 / 3 ( s ) ds c 1 = − 2 1 � ( 9 ρ 4 / 3 + 8 ρ − 8 / 3 ˙ ρ 2 )( s ) ds . c 2 = 1080 V. Kaloshin (the ETH-ITS & U of Maryland) Spectral Rigidity February 7, 2019 25 / 28

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