Geometric Quantization of complex Monge-Ampère operator for certain diffusion flows

28/08/2013
Auteurs : Julien Keller
OAI : oai:www.see.asso.fr:2552:5114
DOI :

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Geometric Quantization of complex Monge-Ampère operator for certain diffusion flows

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– Geometric Quantization of complex Monge-Ampère operator for certain diffusion flows – Julien Keller (Aix-Marseille University) 1 / 57 Kähler metrics 1 Kähler metrics 2 Geometric flows 3 Quantum formalism and intrinsic geometric operators 4 Other related geometries 2 / 57 Kähler metrics Consider M a Kähler manifold M complex manifold : differentiable manifold with atlas whose transition functions are holomorphic ↔ integrable complex structure J ∈ End(TM) A Riemannian metric g on M is hermitian if the scalar product on TM is compatible with J ⇒ a real (1,1)-form ω by ω(X,Y) = g(JX,Y), for all tangent vectors X,Y. Locally ω = √ −1 2 n ∑ i,j=1 hi¯jdzi ∧ d¯zj and ∀p ∈ M, hi¯j(p) is positive definite hermitian matrix If dω = 0, then ω is Kähler Example: the projective space endowed with Fubini-Study metric CPn = ⋃n i=0 Ui, Ui ≃ Cn , ωFS Ui = √ −1 2 ∂ ¯∂ log(∑l≠i zl 2 zi 2 ) 3 / 57 Kähler metrics Consider M a Kähler manifold M complex manifold : differentiable manifold with atlas whose transition functions are holomorphic ↔ integrable complex structure J ∈ End(TM) A Riemannian metric g on M is hermitian if the scalar product on TM is compatible with J ⇒ a real (1,1)-form ω by ω(X,Y) = g(JX,Y), for all tangent vectors X,Y. Locally ω = √ −1 2 n ∑ i,j=1 hi¯jdzi ∧ d¯zj and ∀p ∈ M, hi¯j(p) is positive definite hermitian matrix If dω = 0, then ω is Kähler Example: the projective space endowed with Fubini-Study metric CPn = ⋃n i=0 Ui, Ui ≃ Cn , ωFS Ui = √ −1 2 ∂ ¯∂ log(∑l≠i zl 2 zi 2 ) 4 / 57 Kähler metrics Consider M a Kähler manifold M complex manifold : differentiable manifold with atlas whose transition functions are holomorphic ↔ integrable complex structure J ∈ End(TM) A Riemannian metric g on M is hermitian if the scalar product on TM is compatible with J ⇒ a real (1,1)-form ω by ω(X,Y) = g(JX,Y), for all tangent vectors X,Y. Locally ω = √ −1 2 n ∑ i,j=1 hi¯jdzi ∧ d¯zj and ∀p ∈ M, hi¯j(p) is positive definite hermitian matrix If dω = 0, then ω is Kähler Example: the projective space endowed with Fubini-Study metric CPn = ⋃n i=0 Ui, Ui ≃ Cn , ωFS Ui = √ −1 2 ∂ ¯∂ log(∑l≠i zl 2 zi 2 ) 5 / 57 Kähler metrics Consider M a Kähler manifold M complex manifold : differentiable manifold with atlas whose transition functions are holomorphic ↔ integrable complex structure J ∈ End(TM) A Riemannian metric g on M is hermitian if the scalar product on TM is compatible with J ⇒ a real (1,1)-form ω by ω(X,Y) = g(JX,Y), for all tangent vectors X,Y. Locally ω = √ −1 2 n ∑ i,j=1 hi¯jdzi ∧ d¯zj and ∀p ∈ M, hi¯j(p) is positive definite hermitian matrix If dω = 0, then ω is Kähler Example: the projective space endowed with Fubini-Study metric CPn = ⋃n i=0 Ui, Ui ≃ Cn , ωFS Ui = √ −1 2 ∂ ¯∂ log(∑l≠i zl 2 zi 2 ) 6 / 57 Kähler metrics Consider M a Kähler manifold M complex manifold : differentiable manifold with atlas whose transition functions are holomorphic ↔ integrable complex structure J ∈ End(TM) A Riemannian metric g on M is hermitian if the scalar product on TM is compatible with J ⇒ a real (1,1)-form ω by ω(X,Y) = g(JX,Y), for all tangent vectors X,Y. Locally ω = √ −1 2 n ∑ i,j=1 hi¯jdzi ∧ d¯zj and ∀p ∈ M, hi¯j(p) is positive definite hermitian matrix If dω = 0, then ω is Kähler Example: the projective space endowed with Fubini-Study metric CPn = ⋃n i=0 Ui, Ui ≃ Cn , ωFS Ui = √ −1 2 ∂ ¯∂ log(∑l≠i zl 2 zi 2 ) 7 / 57 Kähler metrics M compact Kähler manifold, n = dimC M, ω Kähler form. Kähler class Ka(ω) = {φ ∈ C∞ (M,R) ω + √ −1∂ ¯∂φ > 0} Theorem (Yau -1978) Let Ω a smooth volume form with ∫M Ω = Vol([ω]). Then there exists a smooth solution φ to the Monge-Ampère equation (ω + √ −1∂ ¯∂φ)n = Ω 8 / 57 Kähler metrics M compact Kähler manifold, n = dimC M, ω Kähler form. Kähler class Ka(ω) = {φ ∈ C∞ (M,R) ω + √ −1∂ ¯∂φ > 0} Theorem (Yau -1978) Let Ω a smooth volume form with ∫M Ω = Vol([ω]). Then there exists a smooth solution φ to the Monge-Ampère equation (ω + √ −1∂ ¯∂φ)n = Ω ↪ Non constructive proof. Transcendental solution 9 / 57 Kähler metrics M compact Kähler manifold, n = dimC M, ω Kähler form. Kähler class Ka(ω) = {φ ∈ C∞ (M,R) ω + √ −1∂ ¯∂φ > 0} Theorem (Yau -1978) Let Ω a smooth volume form with ∫M Ω = Vol([ω]). Then there exists a smooth solution φ to the Monge-Ampère equation (ω + √ −1∂ ¯∂φ)n = Ω ↪ Non constructive proof. Transcendental solution Ricci curvature of ω Ric(ω) = − √ −1∂ ¯∂ log(ωn ) “The Ricci Curvature as organizing principle” Scalar curvature scal(ω) = trace of the Ricci curvature 10 / 57 Kähler metrics Some consequences of Yau’s theorem A new physics (Supersymmetric String Theory) ↔ Ricci flat 3-folds 11 / 57 Kähler metrics Some consequences of Yau’s theorem Smooth probabilities den- sities on M ←→ Space Ka(ω) of Kähler metrics compatible with the symplectic structure ω 12 / 57 Kähler metrics Some consequences of Yau’s theorem Smooth probabilities den- sities on M ←→ Space Ka(ω) of Kähler metrics compatible with the symplectic structure ω Rao-Fisher metric ←→ Calabi metric gF(x,y) µ = ∫M x µ y µµ gC(α,β) ωφ = ∫M ∆ωφ α∆ωφ β ωn φ n! 13 / 57 Kähler metrics Some consequences of Yau’s theorem Smooth probabilities den- sities on M ←→ Space Ka(ω) of Kähler metrics compatible with the symplectic structure ω Rao-Fisher metric ←→ Calabi metric gF(x,y) µ = ∫M x µ y µµ gC(α,β) ωφ = ∫M ∆ωφ α∆ωφ β ωn φ n! gC has constant > 0 sectional curvature. Geodesic equation wrt gC is an ODE ⇒ smoothness and uniqueness 14 / 57 Kähler metrics Some consequences of Yau’s theorem Smooth probabilities den- sities on M ←→ Space Ka(ω) of Kähler metrics compatible with the symplectic structure ω Rao-Fisher metric ←→ Calabi metric gF(x,y) µ = ∫M x µ y µµ gC(α,β) ωφ = ∫M ∆ωφ α∆ωφ β ωn φ n! gC has constant > 0 sectional curvature. Geodesic equation wrt gC is an ODE ⇒ smoothness and uniqueness ↪ works in a more general setup (non compact, singular) 15 / 57 Kähler metrics Radar detection: complex autoregressive model For each echo of the waves sent, amplitude & phase are measured ⇒ Observation values associated to waves are complex vectors Z 16 / 57 Kähler metrics Radar detection: complex autoregressive model For each echo of the waves sent, amplitude & phase are measured ⇒ Observation values associated to waves are complex vectors Z ⇒ Associated (stationary) Gaussian process & covariance matrix E[ZZ∗ ] that is (Toeplitz) hermitian > 0. ↪ Kähler metric (of Bergman type) √ −1∂ ¯∂ logdet(E[ZZ∗ ]), studied by Burbea-Rao (1984) and more recently by F. Barbaresco. 17 / 57 Kähler metrics Radar detection: complex autoregressive model For each echo of the waves sent, amplitude & phase are measured ⇒ Observation values associated to waves are complex vectors Z ⇒ Associated (stationary) Gaussian process & covariance matrix E[ZZ∗ ] that is (Toeplitz) hermitian > 0. ↪ Kähler metric (of Bergman type) √ −1∂ ¯∂ logdet(E[ZZ∗ ]), studied by Burbea-Rao (1984) and more recently by F. Barbaresco. ↪ Kähler metric with constant scalar curvature on bounded homogeneous domains 18 / 57 Kähler metrics Radar detection: complex autoregressive model For each echo of the waves sent, amplitude & phase are measured ⇒ Observation values associated to waves are complex vectors Z ⇒ Associated (stationary) Gaussian process & covariance matrix E[ZZ∗ ] that is (Toeplitz) hermitian > 0. ↪ Kähler metric (of Bergman type) √ −1∂ ¯∂ logdet(E[ZZ∗ ]), studied by Burbea-Rao (1984) and more recently by F. Barbaresco. ↪ Kähler metric with constant scalar curvature on bounded homogeneous domains Open questions for target detection: – define a good distance between two Toeplitz covariance matrices ↔ geodesic distance on Kähler metrics – give a reasonable definition of the average of covariance matrices ↔ balancing/barycenter condition 19 / 57 Kähler metrics Quantum Field Theory Classical system: Phase space (M,ω), observables C∞ (M,R) Quantized system: Hilbert space H(M,ω), hermitian operators on H(M,ω) Quantum phase space P(H): the Fubini-Study metric provides the means of measuring information in quantum mechanics 20 / 57 Kähler metrics Quantum Field Theory Classical system: Phase space (M,ω), observables C∞ (M,R) Quantized system: Hilbert space H(M,ω), hermitian operators on H(M,ω) Quantum phase space P(H): the Fubini-Study metric provides the means of measuring information in quantum mechanics Statistical manifold P⋆ n = {p {x1,...,xn} → R,p(xi) > 0,∑i p(xi) = 1} of non-vanishing probability distributions p on a discrete set {x1,...,xn}. Exponential representation for the tangent space at p ∈ P⋆ n : TpP⋆ n = {u = (u1,...,un) ∈ Rn u1p(x1) + ... + unp(xn) = 0} 21 / 57 Kähler metrics Quantum Field Theory Classical system: Phase space (M,ω), observables C∞ (M,R) Quantized system: Hilbert space H(M,ω), hermitian operators on H(M,ω) Quantum phase space P(H): the Fubini-Study metric provides the means of measuring information in quantum mechanics Statistical manifold P⋆ n = {p {x1,...,xn} → R,p(xi) > 0,∑i p(xi) = 1} of non-vanishing probability distributions p on a discrete set {x1,...,xn}. Exponential representation for the tangent space at p ∈ P⋆ n : TpP⋆ n = {u = (u1,...,un) ∈ Rn u1p(x1) + ... + unp(xn) = 0} ↪ gF Fisher metric, exponential and mixture connections ∇(e) ,∇(m) that are dually flat ⇒ Kähler structure ωF for TP⋆ n . 22 / 57 Kähler metrics Quantum Field Theory Classical system: Phase space (M,ω), observables C∞ (M,R) Quantized system: Hilbert space H(M,ω), hermitian operators on H(M,ω) Quantum phase space P(H): the Fubini-Study metric provides the means of measuring information in quantum mechanics Statistical manifold P⋆ n = {p {x1,...,xn} → R,p(xi) > 0,∑i p(xi) = 1} of non-vanishing probability distributions p on a discrete set {x1,...,xn}. Exponential representation for the tangent space at p ∈ P⋆ n : TpP⋆ n = {u = (u1,...,un) ∈ Rn u1p(x1) + ... + unp(xn) = 0} ↪ gF Fisher metric, exponential and mixture connections ∇(e) ,∇(m) that are dually flat ⇒ Kähler structure ωF for TP⋆ n . Define γ (TP⋆ n ,ωF) → {[z1,..,zn] ∀i, zi ≠ 0} ⊂ (CPn ,ωFS) universal covering map ⇒ local isomorphism of Kähler structures. 23 / 57 Geometric flows 1 Kähler metrics 2 Geometric flows 3 Quantum formalism and intrinsic geometric operators 4 Other related geometries 24 / 57 Geometric flows Deformation of Kähler metrics Intrinsic geometric flows in Riemannian geometry have been used for a few decades (Yamabe flow, Ricci flow, Calabi flow...) for theoretical purposes. 25 / 57 Geometric flows Deformation of Kähler metrics Intrinsic geometric flows in Riemannian geometry have been used for a few decades (Yamabe flow, Ricci flow, Calabi flow...) for theoretical purposes. New applications in recent years Ricci flow in (real) dimension 2: 3D surface shape analysis (shape matching, WP metrics..) 26 / 57 Geometric flows Deformation of Kähler metrics Intrinsic geometric flows in Riemannian geometry have been used for a few decades (Yamabe flow, Ricci flow, Calabi flow...) for theoretical purposes. New applications in recent years Ricci flow in (real) dimension 2: 3D surface shape analysis (shape matching, WP metrics..) ↪ cf. works of X. D. Gu (Stony Brook), G. Zou (Wayne State University), E. Sharon & D. Mumford (Brown University), etc. 27 / 57 Geometric flows Deformation of Kähler metrics Intrinsic geometric flows in Riemannian geometry have been used for a few decades (Yamabe flow, Ricci flow, Calabi flow...) for theoretical purposes. New applications in recent years Ricci flow in (real) dimension 2: 3D surface shape analysis (shape matching, WP metrics..) Ricci/Calabi flow in higher dimension: F. Barbaresco was using Calabi flow in the CAR model 28 / 57 Geometric flows Deformation of Kähler metrics Intrinsic geometric flows in Riemannian geometry have been used for a few decades (Yamabe flow, Ricci flow, Calabi flow...) for theoretical purposes. New applications in recent years Ricci flow in (real) dimension 2: 3D surface shape analysis (shape matching, WP metrics..) Ricci/Calabi flow in higher dimension: F. Barbaresco was using Calabi flow in the CAR model (Normalized) Kähler-Ricci flow ∂ωt ∂t = −Ric(ωt) + λωt, λ ∈ R Kähler Calabi flow ∂φt ∂t = scal(ω + √ −1∂ ¯∂φt) − s These flows may develop singularities ! 29 / 57 Geometric flows Perelman’s functional Boltzmann-Shannon entropy EBS = −∫ M ulogu dV with u(t) = e−f(t) probability density of a particle evolving under Brownian motion ◻∗ u = 0. Fisher information functional, the so-called Perelman’s functional F(g,f) = ∫ M (scal(g) + ∇f 2 )e−f dV since F is the rate of dissipation of entropy: − dEBS dt = F(g,f) 30 / 57 Geometric flows Perelman’s functional Boltzmann-Shannon entropy EBS = −∫ M ulogu dV with u(t) = e−f(t) probability density of a particle evolving under Brownian motion ◻∗ u = 0. Fisher information functional, the so-called Perelman’s functional F(g,f) = ∫ M (scal(g) + ∇f 2 )e−f dV since F is the rate of dissipation of entropy: − dEBS dt = F(g,f) Ricci flow is the gradient flow of F. ↪ Important on Riemannian manifold (Poincaré’s conjecture,...) but also for Kähler manifold (Hamilton-Tian’s conjecture) 31 / 57 Quantum formalism and intrinsic geometric operators 1 Kähler metrics 2 Geometric flows 3 Quantum formalism and intrinsic geometric operators 4 Other related geometries 32 / 57 Quantum formalism and intrinsic geometric operators Berezin Quantization and density of Bergman space Ka(ω) = {φ ∈ C∞ (M,R) ω + √ −1∂ ¯∂φ > 0} ∞-dim Riemannian space. Fix k >> 0, Planck constant ̵h = 1/k, [ω] = c1(L) integral/rational class. Space of Bergman metrics Bk = GL(Nk,C)/U(Nk) set of all hermitian metrics on Hk = H0 (M,L⊗k ), Nk