The hypoelliptic Laplacian

07/11/2017
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The hypoelliptic Laplacian

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The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The hypoelliptic Laplacian Jean-Michel Bismut Université Paris-Sud, Orsay Geometric Science of Information, November 7th 2017 Jean-Michel Bismut The hypoelliptic Laplacian 1 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References 1 The hypoelliptic Laplacian 2 The collapsing of LX b to −∆X /2 3 Dynamical aspects of the interpolation 4 The variational approach 5 The case of S1 6 The limit b → +∞ and the trace formula 7 The hypoelliptic Laplacian and ‘physics’ Jean-Michel Bismut The hypoelliptic Laplacian 2 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References Laplacian and geodesic flow Jean-Michel Bismut The hypoelliptic Laplacian 3 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References Laplacian and geodesic flow X Riemannian manifold of dimension n, X total space of TX, of dimension 2n. Jean-Michel Bismut The hypoelliptic Laplacian 3 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References Laplacian and geodesic flow X Riemannian manifold of dimension n, X total space of TX, of dimension 2n. ∆X Laplace-Beltrami operator acting on X. Jean-Michel Bismut The hypoelliptic Laplacian 3 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References Laplacian and geodesic flow X Riemannian manifold of dimension n, X total space of TX, of dimension 2n. ∆X Laplace-Beltrami operator acting on X. Z geodesic flow: vector field on X. Jean-Michel Bismut The hypoelliptic Laplacian 3 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References Laplacian and geodesic flow X Riemannian manifold of dimension n, X total space of TX, of dimension 2n. ∆X Laplace-Beltrami operator acting on X. Z geodesic flow: vector field on X. Integral trajectories of Z project on X to geodesics ẍ = 0. Jean-Michel Bismut The hypoelliptic Laplacian 3 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The main statement 1 Jean-Michel Bismut The hypoelliptic Laplacian 4 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The main statement • X Riemannian manifold. 1 Jean-Michel Bismut The hypoelliptic Laplacian 4 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The main statement • X Riemannian manifold. • ∆X Laplace-Beltrami, Z generator of geodesic flow. 1 Jean-Michel Bismut The hypoelliptic Laplacian 4 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The main statement • X Riemannian manifold. • ∆X Laplace-Beltrami, Z generator of geodesic flow. The main statement −∆X /2|b=0 LX b |b>0 − − − − − − − − − − − − → −Z|b=+∞. 1 Jean-Michel Bismut The hypoelliptic Laplacian 4 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The main statement • X Riemannian manifold. • ∆X Laplace-Beltrami, Z generator of geodesic flow. The main statement −∆X /2|b=0 LX b |b>0 − − − − − − − − − − − − → −Z|b=+∞. There is a canonical family of hypoelliptic1 operators LX b |b>0 1 almost as good as elliptic. Jean-Michel Bismut The hypoelliptic Laplacian 4 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The main statement • X Riemannian manifold. • ∆X Laplace-Beltrami, Z generator of geodesic flow. The main statement −∆X /2|b=0 LX b |b>0 − − − − − − − − − − − − → −Z|b=+∞. There is a canonical family of hypoelliptic1 operators LX b |b>0 acting on a bigger space X, the total space of TX, 1 almost as good as elliptic. Jean-Michel Bismut The hypoelliptic Laplacian 4 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The main statement • X Riemannian manifold. • ∆X Laplace-Beltrami, Z generator of geodesic flow. The main statement −∆X /2|b=0 LX b |b>0 − − − − − − − − − − − − → −Z|b=+∞. There is a canonical family of hypoelliptic1 operators LX b |b>0 acting on a bigger space X, the total space of TX, that interpolates between −∆X /2 and −Z. 1 almost as good as elliptic. Jean-Michel Bismut The hypoelliptic Laplacian 4 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The dynamical counterpart Jean-Michel Bismut The hypoelliptic Laplacian 5 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The dynamical counterpart For b > 0, b2 ẍ + ẋ = ẇ |{z} white noise Langevin equation. Jean-Michel Bismut The hypoelliptic Laplacian 5 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The dynamical counterpart For b > 0, b2 ẍ + ẋ = ẇ |{z} white noise Langevin equation. There is an interpolation of dynamical systems Jean-Michel Bismut The hypoelliptic Laplacian 5 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The dynamical counterpart For b > 0, b2 ẍ + ẋ = ẇ |{z} white noise Langevin equation. There is an interpolation of dynamical systems ẋ = ẇ | {z } Brownian motion |b=0 b2ẍ+ẋ=ẇ |b>0 − − − − − − − − − − − − → ẍ = 0 | {z } geodesics |b=+∞ . . . Jean-Michel Bismut The hypoelliptic Laplacian 5 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The dynamical counterpart For b > 0, b2 ẍ + ẋ = ẇ |{z} white noise Langevin equation. There is an interpolation of dynamical systems ẋ = ẇ | {z } Brownian motion |b=0 b2ẍ+ẋ=ẇ |b>0 − − − − − − − − − − − − → ẍ = 0 | {z } geodesics |b=+∞ . . . . . . which is the dynamical counterpart to . . . Jean-Michel Bismut The hypoelliptic Laplacian 5 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The dynamical counterpart For b > 0, b2 ẍ + ẋ = ẇ |{z} white noise Langevin equation. There is an interpolation of dynamical systems ẋ = ẇ | {z } Brownian motion |b=0 b2ẍ+ẋ=ẇ |b>0 − − − − − − − − − − − − → ẍ = 0 | {z } geodesics |b=+∞ . . . . . . which is the dynamical counterpart to . . . −∆X /2|b=0 LX b |b>0 − − − − − − − − − − − − → −Z|b=+∞. Jean-Michel Bismut The hypoelliptic Laplacian 5 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References What is the hypoelliptic Laplacian good for? Jean-Michel Bismut The hypoelliptic Laplacian 6 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References What is the hypoelliptic Laplacian good for? 1 The deformation preserves subtle spectral invariants. Jean-Michel Bismut The hypoelliptic Laplacian 6 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References What is the hypoelliptic Laplacian good for? 1 The deformation preserves subtle spectral invariants. 2 On certain manifolds, the deformation is essentially isospectral. Jean-Michel Bismut The hypoelliptic Laplacian 6 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References What is the hypoelliptic Laplacian good for? 1 The deformation preserves subtle spectral invariants. 2 On certain manifolds, the deformation is essentially isospectral. 3 It connects two different kinds of dynamics. Jean-Michel Bismut The hypoelliptic Laplacian 6 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References How does LX b look like? Jean-Michel Bismut The hypoelliptic Laplacian 7 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References How does LX b look like? (x, Y ) ∈ X ⇐⇒ x ∈ X, Y ∈ TxX. Jean-Michel Bismut The hypoelliptic Laplacian 7 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References How does LX b look like? (x, Y ) ∈ X ⇐⇒ x ∈ X, Y ∈ TxX. H = 1 2 −∆TX + |Y |2 − n  harmonic oscillator along the fibre TX. Jean-Michel Bismut The hypoelliptic Laplacian 7 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References How does LX b look like? (x, Y ) ∈ X ⇐⇒ x ∈ X, Y ∈ TxX. H = 1 2 −∆TX + |Y |2 − n  harmonic oscillator along the fibre TX. Z generator of geodesic flow (Z ' Pn i=1 Y i ∂ ∂xi ). Jean-Michel Bismut The hypoelliptic Laplacian 7 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References How does LX b look like? (x, Y ) ∈ X ⇐⇒ x ∈ X, Y ∈ TxX. H = 1 2 −∆TX + |Y |2 − n  harmonic oscillator along the fibre TX. Z generator of geodesic flow (Z ' Pn i=1 Y i ∂ ∂xi ). LX b = H b2 − Z b + . . .. Jean-Michel Bismut The hypoelliptic Laplacian 7 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References How does LX b look like? (x, Y ) ∈ X ⇐⇒ x ∈ X, Y ∈ TxX. H = 1 2 −∆TX + |Y |2 − n  harmonic oscillator along the fibre TX. Z generator of geodesic flow (Z ' Pn i=1 Y i ∂ ∂xi ). LX b = H b2 − Z b + . . .. LX b geometric Fokker-Planck operator (non self-adjoint, nonelliptic, hypoelliptic). Jean-Michel Bismut The hypoelliptic Laplacian 7 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References How is it constructed Jean-Michel Bismut The hypoelliptic Laplacian 8 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References How is it constructed The construction of the hypoelliptic Laplacian is determined by the considered geometries. Jean-Michel Bismut The hypoelliptic Laplacian 8 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References How is it constructed The construction of the hypoelliptic Laplacian is determined by the considered geometries. The hypoelliptic Laplacian is in general non scalar. Jean-Michel Bismut The hypoelliptic Laplacian 8 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References How is it constructed The construction of the hypoelliptic Laplacian is determined by the considered geometries. The hypoelliptic Laplacian is in general non scalar. In the context of de Rham theory, its construction involves the symplectic form on the total space X of T∗ X ' TX. Jean-Michel Bismut The hypoelliptic Laplacian 8 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References A picture Jean-Michel Bismut The hypoelliptic Laplacian 9 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References A picture Jean-Michel Bismut The hypoelliptic Laplacian 9 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The matrix structure of LX b Jean-Michel Bismut The hypoelliptic Laplacian 10 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The matrix structure of LX b X compact Riemannian manifold. Jean-Michel Bismut The hypoelliptic Laplacian 10 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The matrix structure of LX b X compact Riemannian manifold. LX b = H b2 − Z b hypoelliptic Laplacian. Jean-Michel Bismut The hypoelliptic Laplacian 10 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The matrix structure of LX b X compact Riemannian manifold. LX b = H b2 − Z b hypoelliptic Laplacian. H self-adjoint ≥ 0. Jean-Michel Bismut The hypoelliptic Laplacian 10 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The matrix structure of LX b X compact Riemannian manifold. LX b = H b2 − Z b hypoelliptic Laplacian. H self-adjoint ≥ 0. ker H spanned by exp − |Y |2 /2  . Jean-Michel Bismut The hypoelliptic Laplacian 10 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The matrix structure of LX b X compact Riemannian manifold. LX b = H b2 − Z b hypoelliptic Laplacian. H self-adjoint ≥ 0. ker H spanned by exp − |Y |2 /2  . Z ' Pn i=1 Y i ∂ ∂xi maps ker H into ker H⊥ . Jean-Michel Bismut The hypoelliptic Laplacian 10 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The matrix structure of LX b X compact Riemannian manifold. LX b = H b2 − Z b hypoelliptic Laplacian. H self-adjoint ≥ 0. ker H spanned by exp − |Y |2 /2  . Z ' Pn i=1 Y i ∂ ∂xi maps ker H into ker H⊥ . Matrix structure of LX b Jean-Michel Bismut The hypoelliptic Laplacian 10 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The matrix structure of LX b X compact Riemannian manifold. LX b = H b2 − Z b hypoelliptic Laplacian. H self-adjoint ≥ 0. ker H spanned by exp − |Y |2 /2  . Z ' Pn i=1 Y i ∂ ∂xi maps ker H into ker H⊥ . Matrix structure of LX b LX b '  0 −Z/b −Z/b H/b2  . Jean-Michel Bismut The hypoelliptic Laplacian 10 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The matrix structure of LX b X compact Riemannian manifold. LX b = H b2 − Z b hypoelliptic Laplacian. H self-adjoint ≥ 0. ker H spanned by exp − |Y |2 /2  . Z ' Pn i=1 Y i ∂ ∂xi maps ker H into ker H⊥ . Matrix structure of LX b LX b '  0 −Z/b −Z/b H/b2  . Let us pretend LX b finite dimensional matrix. Jean-Michel Bismut The hypoelliptic Laplacian 10 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References LX b collapses to −1 2∆X Jean-Michel Bismut The hypoelliptic Laplacian 11 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References LX b collapses to −1 2∆X Asymptotics of resolvent λ − LX b −1 as b → 0 by Gauss method. Jean-Michel Bismut The hypoelliptic Laplacian 11 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References LX b collapses to −1 2∆X Asymptotics of resolvent λ − LX b −1 as b → 0 by Gauss method. As b → 0, if P orthogonal projector on ker H, Jean-Michel Bismut The hypoelliptic Laplacian 11 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References LX b collapses to −1 2∆X Asymptotics of resolvent λ − LX b −1 as b → 0 by Gauss method. As b → 0, if P orthogonal projector on ker H, λ − LX b −1 '  (λ + PZH−1 ZP) −1 0 0 0  . Jean-Michel Bismut The hypoelliptic Laplacian 11 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References LX b collapses to −1 2∆X Asymptotics of resolvent λ − LX b −1 as b → 0 by Gauss method. As b → 0, if P orthogonal projector on ker H, λ − LX b −1 '  (λ + PZH−1 ZP) −1 0 0 0  . ker H ' C∞ (X, R). Jean-Michel Bismut The hypoelliptic Laplacian 11 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References LX b collapses to −1 2∆X Asymptotics of resolvent λ − LX b −1 as b → 0 by Gauss method. As b → 0, if P orthogonal projector on ker H, λ − LX b −1 '  (λ + PZH−1 ZP) −1 0 0 0  . ker H ' C∞ (X, R). Elementary identity PZH−1 ZP = 1 2 ∆X . Jean-Michel Bismut The hypoelliptic Laplacian 11 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References LX b collapses to −1 2∆X Asymptotics of resolvent λ − LX b −1 as b → 0 by Gauss method. As b → 0, if P orthogonal projector on ker H, λ − LX b −1 '  (λ + PZH−1 ZP) −1 0 0 0  . ker H ' C∞ (X, R). Elementary identity PZH−1 ZP = 1 2 ∆X . λ − LX b −1 → P λ + 1 2 ∆X −1 P by collapsing of X on X (BLebeau 2008). Jean-Michel Bismut The hypoelliptic Laplacian 11 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The convergence of heat kernels as b → 0 Jean-Michel Bismut The hypoelliptic Laplacian 12 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The convergence of heat kernels as b → 0 X compact or noncompact with uniform geometry. Jean-Michel Bismut The hypoelliptic Laplacian 12 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The convergence of heat kernels as b → 0 X compact or noncompact with uniform geometry. Theorem (BLebeau 2008, B2011) Jean-Michel Bismut The hypoelliptic Laplacian 12 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The convergence of heat kernels as b → 0 X compact or noncompact with uniform geometry. Theorem (BLebeau 2008, B2011) Given t > 0, there exist c > 0, C > 0 s.t. for 0 < b ≤ 1, qX b,t ((x, Y ) , (x0 , Y 0 )) ≤ C exp  −c  d2 (x, x0 ) + |Y |2 + |Y 0 | 2  . Jean-Michel Bismut The hypoelliptic Laplacian 12 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The convergence of heat kernels as b → 0 X compact or noncompact with uniform geometry. Theorem (BLebeau 2008, B2011) Given t > 0, there exist c > 0, C > 0 s.t. for 0 < b ≤ 1, qX b,t ((x, Y ) , (x0 , Y 0 )) ≤ C exp  −c  d2 (x, x0 ) + |Y |2 + |Y 0 | 2  . As b → 0, qX b,t ((x, Y ) , (x0 , Y 0 )) → π−n/2 exp  − 1 2  |Y |2 + |Y 0 | 2  pX t (x, x0 ) . Jean-Michel Bismut The hypoelliptic Laplacian 12 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The geometric Langevin process Jean-Michel Bismut The hypoelliptic Laplacian 13 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The geometric Langevin process Stochastic process corresponding to the hypoelliptic Laplacian is a geometric Langevin process. Jean-Michel Bismut The hypoelliptic Laplacian 13 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The geometric Langevin process Stochastic process corresponding to the hypoelliptic Laplacian is a geometric Langevin process. LX b = 1 2b2 −∆V + 2∇V Y  − Z b . Jean-Michel Bismut The hypoelliptic Laplacian 13 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The geometric Langevin process Stochastic process corresponding to the hypoelliptic Laplacian is a geometric Langevin process. LX b = 1 2b2 −∆V + 2∇V Y  − Z b . ẋ = Y b , Ẏ = −Y b2 + ẇ b . . . Jean-Michel Bismut The hypoelliptic Laplacian 13 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The geometric Langevin process Stochastic process corresponding to the hypoelliptic Laplacian is a geometric Langevin process. LX b = 1 2b2 −∆V + 2∇V Y  − Z b . ẋ = Y b , Ẏ = −Y b2 + ẇ b . . . . . . projects to Langevin equation b2 ẍ = −ẋ + ẇ. Jean-Michel Bismut The hypoelliptic Laplacian 13 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The geometric Langevin process Stochastic process corresponding to the hypoelliptic Laplacian is a geometric Langevin process. LX b = 1 2b2 −∆V + 2∇V Y  − Z b . ẋ = Y b , Ẏ = −Y b2 + ẇ b . . . . . . projects to Langevin equation b2 ẍ = −ẋ + ẇ. bẋ is a Ornstein-Uhlenbeck process. Jean-Michel Bismut The hypoelliptic Laplacian 13 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The geometric Langevin process Stochastic process corresponding to the hypoelliptic Laplacian is a geometric Langevin process. LX b = 1 2b2 −∆V + 2∇V Y  − Z b . ẋ = Y b , Ẏ = −Y b2 + ẇ b . . . . . . projects to Langevin equation b2 ẍ = −ẋ + ẇ. bẋ is a Ornstein-Uhlenbeck process. When b = 0, we get ẋ = ẇ Brownian motion. Jean-Michel Bismut The hypoelliptic Laplacian 13 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The geometric Langevin process Stochastic process corresponding to the hypoelliptic Laplacian is a geometric Langevin process. LX b = 1 2b2 −∆V + 2∇V Y  − Z b . ẋ = Y b , Ẏ = −Y b2 + ẇ b . . . . . . projects to Langevin equation b2 ẍ = −ẋ + ẇ. bẋ is a Ornstein-Uhlenbeck process. When b = 0, we get ẋ = ẇ Brownian motion. When b = +∞, ẍ = 0 geodesic. Jean-Michel Bismut The hypoelliptic Laplacian 13 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The geometric Langevin process Stochastic process corresponding to the hypoelliptic Laplacian is a geometric Langevin process. LX b = 1 2b2 −∆V + 2∇V Y  − Z b . ẋ = Y b , Ẏ = −Y b2 + ẇ b . . . . . . projects to Langevin equation b2 ẍ = −ẋ + ẇ. bẋ is a Ornstein-Uhlenbeck process. When b = 0, we get ẋ = ẇ Brownian motion. When b = +∞, ẍ = 0 geodesic. From an algebraic point of view, interpolation property is obvious. Jean-Michel Bismut The hypoelliptic Laplacian 13 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The geometric process (x·, Y·) as b → 0 Jean-Michel Bismut The hypoelliptic Laplacian 14 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The geometric process (x·, Y·) as b → 0 Theorem (B 2011) As b → 0: Jean-Michel Bismut The hypoelliptic Laplacian 14 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The geometric process (x·, Y·) as b → 0 Theorem (B 2011) As b → 0: The law of x· converges to Brownian motion. Jean-Michel Bismut The hypoelliptic Laplacian 14 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The geometric process (x·, Y·) as b → 0 Theorem (B 2011) As b → 0: The law of x· converges to Brownian motion. If 0 < t1 < t2 . . . < tm, (bẋt1 , . . . , bẋtm ) converges to product of independent Gaussians with σ2 = 1/2. Jean-Michel Bismut The hypoelliptic Laplacian 14 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The geometric process (x·, Y·) as b → 0 Theorem (B 2011) As b → 0: The law of x· converges to Brownian motion. If 0 < t1 < t2 . . . < tm, (bẋt1 , . . . , bẋtm ) converges to product of independent Gaussians with σ2 = 1/2. Law of (x·, bẋt1 , . . . , bẋtm ) converges to the product law. Jean-Michel Bismut The hypoelliptic Laplacian 14 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The geometric process (x·, Y·) as b → 0 Theorem (B 2011) As b → 0: The law of x· converges to Brownian motion. If 0 < t1 < t2 . . . < tm, (bẋt1 , . . . , bẋtm ) converges to product of independent Gaussians with σ2 = 1/2. Law of (x·, bẋt1 , . . . , bẋtm ) converges to the product law. R t 0 b2 |ẋ|2 ds → t/2. Jean-Michel Bismut The hypoelliptic Laplacian 14 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The geometric process (x·, Y·) as b → 0 Theorem (B 2011) As b → 0: The law of x· converges to Brownian motion. If 0 < t1 < t2 . . . < tm, (bẋt1 , . . . , bẋtm ) converges to product of independent Gaussians with σ2 = 1/2. Law of (x·, bẋt1 , . . . , bẋtm ) converges to the product law. R t 0 b2 |ẋ|2 ds → t/2. Remark This is a geometric form of homogenization. Jean-Michel Bismut The hypoelliptic Laplacian 14 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The action functional Jean-Michel Bismut The hypoelliptic Laplacian 15 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The action functional s ∈ R/tZ → xs ∈ X smooth loop. Jean-Michel Bismut The hypoelliptic Laplacian 15 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The action functional s ∈ R/tZ → xs ∈ X smooth loop. Action functional Ib,t (x) = 1 2 Z R/tZ b2 ẍ + ẋ 2 ds = 1 2 Z R/tZ |ẋ|2 + b4 |ẍ|2 ds. Jean-Michel Bismut The hypoelliptic Laplacian 15 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The action functional s ∈ R/tZ → xs ∈ X smooth loop. Action functional Ib,t (x) = 1 2 Z R/tZ b2 ẍ + ẋ 2 ds = 1 2 Z R/tZ |ẋ|2 + b4 |ẍ|2 ds. For b = 0, Ib,t is the energy Et. Jean-Michel Bismut The hypoelliptic Laplacian 15 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The action functional s ∈ R/tZ → xs ∈ X smooth loop. Action functional Ib,t (x) = 1 2 Z R/tZ b2 ẍ + ẋ 2 ds = 1 2 Z R/tZ |ẋ|2 + b4 |ẍ|2 ds. For b = 0, Ib,t is the energy Et. dQb,t (x) = exp (−Ib,t (x)) Dx. Jean-Michel Bismut The hypoelliptic Laplacian 15 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The action functional s ∈ R/tZ → xs ∈ X smooth loop. Action functional Ib,t (x) = 1 2 Z R/tZ b2 ẍ + ẋ 2 ds = 1 2 Z R/tZ |ẋ|2 + b4 |ẍ|2 ds. For b = 0, Ib,t is the energy Et. dQb,t (x) = exp (−Ib,t (x)) Dx. For b = 0, this is just the Brownian measure dPt. Jean-Michel Bismut The hypoelliptic Laplacian 15 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The action functional s ∈ R/tZ → xs ∈ X smooth loop. Action functional Ib,t (x) = 1 2 Z R/tZ b2 ẍ + ẋ 2 ds = 1 2 Z R/tZ |ẋ|2 + b4 |ẍ|2 ds. For b = 0, Ib,t is the energy Et. dQb,t (x) = exp (−Ib,t (x)) Dx. For b = 0, this is just the Brownian measure dPt. As b → +∞, dQb,t concentrates on ẍ = 0. Jean-Michel Bismut The hypoelliptic Laplacian 15 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References Pseudodistance Jean-Michel Bismut The hypoelliptic Laplacian 16 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References Pseudodistance Ib,t (x) = 1 2 Z [0,t] |ẋ|2 + b4 |ẍ|2 ds. Jean-Michel Bismut The hypoelliptic Laplacian 16 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References Pseudodistance Ib,t (x) = 1 2 Z [0,t] |ẋ|2 + b4 |ẍ|2 ds. Fix (x, ẋ)0 = (x, Y/b) , (x, ẋ)t = (x0 , Y 0 /b). Jean-Michel Bismut The hypoelliptic Laplacian 16 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References Pseudodistance Ib,t (x) = 1 2 Z [0,t] |ẋ|2 + b4 |ẍ|2 ds. Fix (x, ẋ)0 = (x, Y/b) , (x, ẋ)t = (x0 , Y 0 /b). Hb,t ((x, Y ) , (x0 , Y 0 )) inf of Ib,t (x). Jean-Michel Bismut The hypoelliptic Laplacian 16 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References Pseudodistance Ib,t (x) = 1 2 Z [0,t] |ẋ|2 + b4 |ẍ|2 ds. Fix (x, ẋ)0 = (x, Y/b) , (x, ẋ)t = (x0 , Y 0 /b). Hb,t ((x, Y ) , (x0 , Y 0 )) inf of Ib,t (x). In general, Hb,t ((x, Y ) , (x, Y )) > 0. Jean-Michel Bismut The hypoelliptic Laplacian 16 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The case of X = Rn Jean-Michel Bismut The hypoelliptic Laplacian 17 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The case of X = Rn Proposition Jean-Michel Bismut The hypoelliptic Laplacian 17 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The case of X = Rn Proposition On Rn , by explicit computations, as b → 0, Jean-Michel Bismut The hypoelliptic Laplacian 17 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The case of X = Rn Proposition On Rn , by explicit computations, as b → 0, pb,t ((x, Y ) , (x0 , Y 0 )) = Cb,t exp (−Hb,t ((x, Y ) , (x0 , Y 0 ))) . Jean-Michel Bismut The hypoelliptic Laplacian 17 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The case of X = Rn Proposition On Rn , by explicit computations, as b → 0, pb,t ((x, Y ) , (x0 , Y 0 )) = Cb,t exp (−Hb,t ((x, Y ) , (x0 , Y 0 ))) . Moreover, Hb,t ((x, Y ) , (x0 , Y 0 )) → H0,t ((x, Y ) , (x0 , Y 0 )) = 1 2  |Y |2 + |Y 0 | 2  + 1 2t |x0 − x| 2 . Jean-Michel Bismut The hypoelliptic Laplacian 17 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The harmonic oscillator Jean-Michel Bismut The hypoelliptic Laplacian 18 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The harmonic oscillator H = 1 2  − ∂2 ∂y2 + y2 − 1  harmonic oscillator. Jean-Michel Bismut The hypoelliptic Laplacian 18 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The harmonic oscillator H = 1 2  − ∂2 ∂y2 + y2 − 1  harmonic oscillator. H self-adjoint elliptic, Sp (H) = N. Jean-Michel Bismut The hypoelliptic Laplacian 18 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The harmonic oscillator H = 1 2  − ∂2 ∂y2 + y2 − 1  harmonic oscillator. H self-adjoint elliptic, Sp (H) = N. ker H spanned by e−y2/2 . Jean-Michel Bismut The hypoelliptic Laplacian 18 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The harmonic oscillator H = 1 2  − ∂2 ∂y2 + y2 − 1  harmonic oscillator. H self-adjoint elliptic, Sp (H) = N. ker H spanned by e−y2/2 . Eigenfunctions e−y2/2 Pk (y) |k∈N, with Pk Hermite polynomials, are analytic. Jean-Michel Bismut The hypoelliptic Laplacian 18 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References A formal translation Jean-Michel Bismut The hypoelliptic Laplacian 19 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References A formal translation Lb = 1 2b2  − ∂2 ∂y2 + y2 − 1  − y b ∂ ∂x . Jean-Michel Bismut The hypoelliptic Laplacian 19 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References A formal translation Lb = 1 2b2  − ∂2 ∂y2 + y2 − 1  − y b ∂ ∂x . Lb = 1 2b2  − ∂2 ∂y2 + y − b ∂ ∂x 2 − 1  − 1 2 ∂2 ∂x2 . Jean-Michel Bismut The hypoelliptic Laplacian 19 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References A formal translation Lb = 1 2b2  − ∂2 ∂y2 + y2 − 1  − y b ∂ ∂x . Lb = 1 2b2  − ∂2 ∂y2 + y − b ∂ ∂x 2 − 1  − 1 2 ∂2 ∂x2 . Make formal translation y → y + b ∂ ∂x . Jean-Michel Bismut The hypoelliptic Laplacian 19 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References A formal translation Lb = 1 2b2  − ∂2 ∂y2 + y2 − 1  − y b ∂ ∂x . Lb = 1 2b2  − ∂2 ∂y2 + y − b ∂ ∂x 2 − 1  − 1 2 ∂2 ∂x2 . Make formal translation y → y + b ∂ ∂x . Translation ' conjugation. Jean-Michel Bismut The hypoelliptic Laplacian 19 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References A conjugation of Lb Jean-Michel Bismut The hypoelliptic Laplacian 20 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References A conjugation of Lb M = ∂2 ∂x∂y hyperbolic, ebM is not well-defined. Jean-Michel Bismut The hypoelliptic Laplacian 20 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References A conjugation of Lb M = ∂2 ∂x∂y hyperbolic, ebM is not well-defined. Conjugation identity ebM Lbe−bM = 1 2b2  − ∂2 ∂y2 + y2 − 1  − 1 2 ∂2 ∂x2 . Jean-Michel Bismut The hypoelliptic Laplacian 20 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References A conjugation of Lb M = ∂2 ∂x∂y hyperbolic, ebM is not well-defined. Conjugation identity ebM Lbe−bM = 1 2b2  − ∂2 ∂y2 + y2 − 1  − 1 2 ∂2 ∂x2 . Lb hypoelliptic, Jean-Michel Bismut The hypoelliptic Laplacian 20 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References A conjugation of Lb M = ∂2 ∂x∂y hyperbolic, ebM is not well-defined. Conjugation identity ebM Lbe−bM = 1 2b2  − ∂2 ∂y2 + y2 − 1  − 1 2 ∂2 ∂x2 . Lb hypoelliptic, ebM Lbe−bM elliptic. Jean-Michel Bismut The hypoelliptic Laplacian 20 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References A conjugation of Lb M = ∂2 ∂x∂y hyperbolic, ebM is not well-defined. Conjugation identity ebM Lbe−bM = 1 2b2  − ∂2 ∂y2 + y2 − 1  − 1 2 ∂2 ∂x2 . Lb hypoelliptic, ebM Lbe−bM elliptic. If conjugation can be made sense of, the two operators are isospectral. Jean-Michel Bismut The hypoelliptic Laplacian 20 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References Conjugation is legitimate Jean-Michel Bismut The hypoelliptic Laplacian 21 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References Conjugation is legitimate By analyticity, y → y + ibξ acts the eigenfunctions of H e−y2/2 Pk (y). Jean-Michel Bismut The hypoelliptic Laplacian 21 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References Conjugation is legitimate By analyticity, y → y + ibξ acts the eigenfunctions of H e−y2/2 Pk (y). Lb can be explicitly diagonalized . . . Jean-Michel Bismut The hypoelliptic Laplacian 21 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References Conjugation is legitimate By analyticity, y → y + ibξ acts the eigenfunctions of H e−y2/2 Pk (y). Lb can be explicitly diagonalized . . . . . . on a very distorted base. Jean-Michel Bismut The hypoelliptic Laplacian 21 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The spectrum of Lb Jean-Michel Bismut The hypoelliptic Laplacian 22 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The spectrum of Lb Lb = 1 2b2  − ∂2 ∂y2 + y2 − 1  − y b ∂ ∂x . Jean-Michel Bismut The hypoelliptic Laplacian 22 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The spectrum of Lb Lb = 1 2b2  − ∂2 ∂y2 + y2 − 1  − y b ∂ ∂x . Sp (Lb) = {2k2 π2 , k ∈ Z} + N b2 . Jean-Michel Bismut The hypoelliptic Laplacian 22 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The spectrum of Lb Lb = 1 2b2  − ∂2 ∂y2 + y2 − 1  − y b ∂ ∂x . Sp (Lb) = {2k2 π2 , k ∈ Z} + N b2 . Spectrum of −∆S1 /2 remains rigidly embedded in Sp (Lb). Jean-Michel Bismut The hypoelliptic Laplacian 22 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The spectrum of Lb Lb = 1 2b2  − ∂2 ∂y2 + y2 − 1  − y b ∂ ∂x . Sp (Lb) = {2k2 π2 , k ∈ Z} + N b2 . Spectrum of −∆S1 /2 remains rigidly embedded in Sp (Lb). When b → 0, Sp (Lb) → Sp  −∆S1 /2  . Jean-Michel Bismut The hypoelliptic Laplacian 22 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The operator Lb as b → +∞ Jean-Michel Bismut The hypoelliptic Laplacian 23 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The operator Lb as b → +∞ When b → +∞, Lb ' 1 2 y2 − y ∂ ∂x . . . Jean-Michel Bismut The hypoelliptic Laplacian 23 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The operator Lb as b → +∞ When b → +∞, Lb ' 1 2 y2 − y ∂ ∂x . . . When b → +∞, heat propagates more and more along trajectories of y ∂ ∂x (geodesics in S1 ). Jean-Michel Bismut The hypoelliptic Laplacian 23 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References Supersymmetry and Poisson formula Jean-Michel Bismut The hypoelliptic Laplacian 24 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References Supersymmetry and Poisson formula NΛ·(R) degree counting operator on Λ· (R) = R ⊕ R. Jean-Michel Bismut The hypoelliptic Laplacian 24 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References Supersymmetry and Poisson formula NΛ·(R) degree counting operator on Λ· (R) = R ⊕ R. Lb = Lb + NΛ·(R) b2 has the same spectrum as Lb. Jean-Michel Bismut The hypoelliptic Laplacian 24 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References Supersymmetry and Poisson formula NΛ·(R) degree counting operator on Λ· (R) = R ⊕ R. Lb = Lb + NΛ·(R) b2 has the same spectrum as Lb. Tr h exp  t∆S1 /2 i = Trs |{z} difference of traces [exp (−tLb)]. Jean-Michel Bismut The hypoelliptic Laplacian 24 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References Supersymmetry and Poisson formula NΛ·(R) degree counting operator on Λ· (R) = R ⊕ R. Lb = Lb + NΛ·(R) b2 has the same spectrum as Lb. Tr h exp  t∆S1 /2 i = Trs |{z} difference of traces [exp (−tLb)]. When b → +∞, Poisson formula follows by interpolation. Jean-Michel Bismut The hypoelliptic Laplacian 24 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References Supersymmetry and Poisson formula NΛ·(R) degree counting operator on Λ· (R) = R ⊕ R. Lb = Lb + NΛ·(R) b2 has the same spectrum as Lb. Tr h exp  t∆S1 /2 i = Trs |{z} difference of traces [exp (−tLb)]. When b → +∞, Poisson formula follows by interpolation. The ‘natural’ geometric deformation Lb of −∆S1 /2 is a nonscalar operator. Jean-Michel Bismut The hypoelliptic Laplacian 24 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References Selberg’s trace formula Jean-Michel Bismut The hypoelliptic Laplacian 25 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References Selberg’s trace formula X Riemann surface of constant negative curvature. Jean-Michel Bismut The hypoelliptic Laplacian 25 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References Selberg’s trace formula X Riemann surface of constant negative curvature. lγ length of closed geodesics γ. Jean-Michel Bismut The hypoelliptic Laplacian 25 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References Selberg’s trace formula X Riemann surface of constant negative curvature. lγ length of closed geodesics γ. Tr  exp t∆X /2  | {z } Laplacian = exp (−t/8) 2πt Vol (Σ) | {z } geodesic flow Jean-Michel Bismut The hypoelliptic Laplacian 25 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References Selberg’s trace formula X Riemann surface of constant negative curvature. lγ length of closed geodesics γ. Tr  exp t∆X /2  | {z } Laplacian = exp (−t/8) 2πt Vol (Σ) | {z } geodesic flow Z R exp −y2 /2t  y/2 sinh (y/2) dy √ 2πt Jean-Michel Bismut The hypoelliptic Laplacian 25 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References Selberg’s trace formula X Riemann surface of constant negative curvature. lγ length of closed geodesics γ. Tr  exp t∆X /2  | {z } Laplacian = exp (−t/8) 2πt Vol (Σ) | {z } geodesic flow Z R exp −y2 /2t  y/2 sinh (y/2) dy √ 2πt + X γ6=0 Volγ √ 2πt exp −`2 γ/2t − t/8  2 sinh (`γ/2) . Jean-Michel Bismut The hypoelliptic Laplacian 25 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References A proof via the hypoelliptic Laplacian Jean-Michel Bismut The hypoelliptic Laplacian 26 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References A proof via the hypoelliptic Laplacian On the total space b X of a vector bundle TX ⊕ N of dimension 3 over X. . . Jean-Michel Bismut The hypoelliptic Laplacian 26 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References A proof via the hypoelliptic Laplacian On the total space b X of a vector bundle TX ⊕ N of dimension 3 over X. . . . . . one constructs a hypoelliptic Laplacian LX b such that as in the case of S1 . . . Jean-Michel Bismut The hypoelliptic Laplacian 26 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References A proof via the hypoelliptic Laplacian On the total space b X of a vector bundle TX ⊕ N of dimension 3 over X. . . . . . one constructs a hypoelliptic Laplacian LX b such that as in the case of S1 . . . Tr  exp t∆X /2  = Trs  exp −tLX b  . Jean-Michel Bismut The hypoelliptic Laplacian 26 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References A proof via the hypoelliptic Laplacian On the total space b X of a vector bundle TX ⊕ N of dimension 3 over X. . . . . . one constructs a hypoelliptic Laplacian LX b such that as in the case of S1 . . . Tr  exp t∆X /2  = Trs  exp −tLX b  . Make b → +∞. Jean-Michel Bismut The hypoelliptic Laplacian 26 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References A proof via the hypoelliptic Laplacian On the total space b X of a vector bundle TX ⊕ N of dimension 3 over X. . . . . . one constructs a hypoelliptic Laplacian LX b such that as in the case of S1 . . . Tr  exp t∆X /2  = Trs  exp −tLX b  . Make b → +∞. One obtains Selberg like formulas for all compact locally symmetric spaces (B2011). Jean-Michel Bismut The hypoelliptic Laplacian 26 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References A formula for LX b Jean-Michel Bismut The hypoelliptic Laplacian 27 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References A formula for LX b LX b = 1 2  Y N , Y TX  2 + 1 2b2 −∆TX⊕N + |Y |2 − n  + NΛ·(T∗X⊕N∗) b2 Jean-Michel Bismut The hypoelliptic Laplacian 27 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References A formula for LX b LX b = 1 2  Y N , Y TX  2 + 1 2b2 −∆TX⊕N + |Y |2 − n  + NΛ·(T∗X⊕N∗) b2 + 1 b ∇Y T X +b c ad Y TX  −c ad Y TX  + iθad Y N  ! . Jean-Michel Bismut The hypoelliptic Laplacian 27 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References A formula for LX b LX b = 1 2  Y N , Y TX  2 + 1 2b2 −∆TX⊕N + |Y |2 − n  + NΛ·(T∗X⊕N∗) b2 + 1 b ∇Y T X +b c ad Y TX  −c ad Y TX  + iθad Y N  ! . LX b is a deformation of 1 2 −∆X + c  . Jean-Michel Bismut The hypoelliptic Laplacian 27 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The harmonic oscillator Jean-Michel Bismut The hypoelliptic Laplacian 28 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The harmonic oscillator Figure: Fibrewise gymnastics Jean-Michel Bismut The hypoelliptic Laplacian 28 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The elliptical machine Jean-Michel Bismut The hypoelliptic Laplacian 29 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The elliptical machine Figure: Fibrewise gymnastics Jean-Michel Bismut The hypoelliptic Laplacian 29 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The Langevin equation Jean-Michel Bismut The hypoelliptic Laplacian 30 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The Langevin equation Langevin tried to reconcile Newton’s law with Brownian motion (infinite energy). Jean-Michel Bismut The hypoelliptic Laplacian 30 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The Langevin equation Langevin tried to reconcile Newton’s law with Brownian motion (infinite energy). Langevin equation mẍ = −ẋ + ẇ in R3 . Jean-Michel Bismut The hypoelliptic Laplacian 30 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The Langevin equation Langevin tried to reconcile Newton’s law with Brownian motion (infinite energy). Langevin equation mẍ = −ẋ + ẇ in R3 . For m = 0, ẋ = ẇ, for m = +∞, ẍ = 0. Jean-Michel Bismut The hypoelliptic Laplacian 30 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References The Langevin equation Langevin tried to reconcile Newton’s law with Brownian motion (infinite energy). Langevin equation mẍ = −ẋ + ẇ in R3 . For m = 0, ẋ = ẇ, for m = +∞, ẍ = 0. If we make m = b2 , the hypoelliptic Laplacian (with Y = ẋ) describes the dynamics of the Langevin equation in a geometric context. Jean-Michel Bismut The hypoelliptic Laplacian 30 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References Langevin (C.R. de l’Académie des Sciences 1908) Jean-Michel Bismut The hypoelliptic Laplacian 31 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References Langevin (C.R. de l’Académie des Sciences 1908) Jean-Michel Bismut The hypoelliptic Laplacian 31 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References Welcome to the role of mass in classical math ! Jean-Michel Bismut The hypoelliptic Laplacian 32 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References J.-M. Bismut and G. Lebeau, The hypoelliptic Laplacian and Ray-Singer metrics, Annals of Mathematics Studies, vol. 167, Princeton University Press, Princeton, NJ, 2008. MR MR2441523 J.-M. Bismut, Loop spaces and the hypoelliptic Laplacian, Comm. Pure Appl. Math. 61 (2008), no. 4, 559–593. MR MR2383933 , Hypoelliptic Laplacian and orbital integrals, Annals of Mathematics Studies, vol. 177, Princeton University Press, Princeton, NJ, 2011. MR 2828080 Jean-Michel Bismut The hypoelliptic Laplacian 33 / 34 The hypoelliptic Laplacian The collapsing of LX b to −∆X /2 Dynamical aspects of the interpolation The variational approach The case of S1 The limit b → +∞ and the trace formula The hypoelliptic Laplacian and ‘physics’ References , Hypoelliptic Laplacian and probability, J. Math. Soc. Japan 67 (2015), no. 4, 1317–1357. MR 3417500 Jean-Michel Bismut The hypoelliptic Laplacian 34 / 34