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Hypoelliptic Laplacian and orbital integrals /
Annotation
| Clasificación: | Libro Electrónico |
|---|---|
| Autor principal: | |
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
Princeton, N.J. :
Princeton University Press,
©2011.
|
| Colección: | Annals of mathematics studies ;
no. 177. |
| Temas: | |
| Acceso en línea: | Texto completo |
MARC
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| 100 | 1 | |a Bismut, Jean-Michel. | |
| 245 | 1 | 0 | |a Hypoelliptic Laplacian and orbital integrals / |c Jean-Michel Bismut. |
| 260 | |a Princeton, N.J. : |b Princeton University Press, |c ©2011. | ||
| 300 | |a 1 online resource (x, 330 pages) | ||
| 336 | |a text |b txt |2 rdacontent | ||
| 337 | |a computer |b c |2 rdamedia | ||
| 338 | |a online resource |b cr |2 rdacarrier | ||
| 347 | |a data file |2 rda | ||
| 490 | 1 | |a Annals of mathematics studies ; |v no. 177 | |
| 504 | |a Includes bibliographical references and indexes. | ||
| 520 | 8 | |a Annotation |b This book uses the hypoelliptic Laplacian to evaluate semisimple orbital integrals in a formalism that unifies index theory and the trace formula. The hypoelliptic Laplacian is a family of operators that is supposed to interpolate between the ordinary Laplacian and the geodesic flow. It is essentially the weighted sum of a harmonic oscillator along the fiber of the tangent bundle, and of the generator of the geodesic flow. In this book, semisimple orbital integrals associated with the heat kernel of the Casimir operator are shown to be invariant under a suitable hypoelliptic deformation, which is constructed using the Dirac operator of Kostant. Their explicit evaluation is obtained by localization on geodesics in the symmetric space, in a formula closely related to the Atiyah-Bott fixed point formulas. Orbital integrals associated with the wave kernel are also computed. Estimates on the hypoelliptic heat kernel play a key role in the proofs, and are obtained by combining analytic, geometric, and probabilistic techniques. Analytic techniques emphasize the wavelike aspects of the hypoelliptic heat kernel, while geometrical considerations are needed to obtain proper control of the hypoelliptic heat kernel, especially in the localization process near the geodesics. Probabilistic techniques are especially relevant, because underlying the hypoelliptic deformation is a deformation of dynamical systems on the symmetric space, which interpolates between Brownian motion and the geodesic flow. | |
| 588 | 0 | |a Print version record. | |
| 505 | 0 | |6 880-01 |a Introduction -- 1. Clifford and Heisenberg algebras -- 2. The hypoelliptic Laplacian onX=G/K -- 3. | |
| 546 | |a English. | ||
| 590 | |a JSTOR |b Books at JSTOR Demand Driven Acquisitions (DDA) | ||
| 590 | |a ProQuest Ebook Central |b Ebook Central Academic Complete | ||
| 650 | 0 | |a Differential equations, Hypoelliptic. | |
| 650 | 0 | |a Laplacian operator. | |
| 650 | 0 | |a Definite integrals. | |
| 650 | 0 | |a Orbit method. | |
| 650 | 6 | |a Équations différentielles hypo-elliptiques. | |
| 650 | 6 | |a Laplacien. | |
| 650 | 6 | |a Intégrales définies. | |
| 650 | 6 | |a Méthode des orbites. | |
| 650 | 7 | |a MATHEMATICS |x Differential Equations |x Partial. |2 bisacsh | |
| 650 | 7 | |a MATHEMATICS |x Mathematical Analysis. |2 bisacsh | |
| 650 | 7 | |a Integrales |2 embne | |
| 650 | 7 | |a Definite integrals |2 fast | |
| 650 | 7 | |a Differential equations, Hypoelliptic |2 fast | |
| 650 | 7 | |a Laplacian operator |2 fast | |
| 650 | 7 | |a Orbit method |2 fast | |
| 758 | |i has work: |a Hypoelliptic Laplacian and orbital integrals (Text) |1 https://id.oclc.org/worldcat/entity/E39PCH74m7Qq6BDPHGWmJtYMRq |4 https://id.oclc.org/worldcat/ontology/hasWork | ||
| 776 | 0 | 8 | |i Print version: |a Bismut, Jean-Michel. |t Hypoelliptic Laplacian and orbital integrals. |d Princeton, N.J. : Princeton University Press, ©2011 |z 9781400840571 |w (OCoLC)751685247 |
| 830 | 0 | |a Annals of mathematics studies ; |v no. 177. | |
| 856 | 4 | 0 | |u https://ebookcentral.uam.elogim.com/lib/uam-ebooks/detail.action?docID=729954 |z Texto completo |
| 880 | 0 | 0 | |6 505-01/(S |g Machine generated contents note: |g 0.1. |t trace formula as a Lefschetz formula -- |g 0.2. |t short history of the hypoelliptic Laplacian -- |g 0.3. |t hypoelliptic Laplacian on a symmetric space -- |g 0.4. |t hypoelliptic Laplacian and its heat kernel -- |g 0.5. |t Elliptic and hypoelliptic orbital integrals -- |g 0.6. |t limit as b -> 0 -- |g 0.7. |t limit as b -> + infinity: an explicit formula for the orbital integrals -- |g 0.8. |t analysis of the hypoelliptic orbital integrals -- |g 0.9. |t heat kernel for bounded b and the Malliavin calculus -- |g 0.10. |t heat kernel for large b, Toponogov, and local index -- |g 0.11. |t hypoelliptic Laplacian and the wave equation -- |g 0.12. |t organization of the book -- |g 1.1. |t Clifford algebra of a real vector space -- |g 1.2. |t Clifford algebra of V direct sum V* -- |g 1.3. |t Heisenberg algebra -- |g 1.4. |t Heisenberg algebra of V direct sum V* -- |g 1.5. |t Clifford-Heisenberg algebra of V direct sum V* -- |g 1.6. |t Clifford-Heisenberg algebra of V direct sum V* when V is Euclidean -- |g 2.1. |t pair (G, K) -- |g 2.2. |t flat connection on TX direct sum N -- |g 2.3. |t Clifford algebras of g -- |g 2.4. |t flat connections on Λ (T* X Edirect sum N*) -- |g 2.5. |t Casimir operator -- |g 2.6. |t form Kg -- |g 2.7. |t Dirac operator of Kostant -- |g 2.8. |t Clifford-Heisenberg algebra of g direct sum g* -- |g 2.9. |t operator Db -- |g 2.10. |t compression of the operator Db -- |g 2.11. |t formula for D2b -- |g 2.12. |t action of Db on quotients by K -- |g 2.13. |t operators LX and Lxb -- |g 2.14. |t scaling of the form B -- |g 2.15. |t Bianchi identity -- |g 2.16. |t fundamental identity -- |g 2.17. |t canonical vector fields on Lxb -- |g 2.18. |t Lie derivatives and the operator a -- |g 3.1. |t Convexity, the displacement function, and its critical set -- |g 3.2. |t norm of the canonical vector fields -- |g 3.3. |t subset X (γ) as a symmetric space -- |g 3.4. |t normal coordinate system on X based at X (γ) -- |g 3.5. |t return map along the minimizing geodesics in X (γ) -- |g 3.6. |t return map on X -- |g 3.7. |t connection form in the parallel transport trivialization -- |g 3.8. |t Distances and pseudodistances on X and X -- |g 3.9. |t pseudodistance and Toponogov's theorem -- |g 3.10. |t flat bundle (TX direct sum N) (γ) -- |g 4.1. |t algebra of invariant kernels on X -- |g 4.2. |t Orbital integrals -- |g 4.3. |t Infinite dimensional orbital integrals -- |g 4.4. |t orbital integrals for the elliptic heat kernel of X -- |g 4.5. |t orbital supertraces for the hypoelliptic heat kernel -- |g 4.6. |t fundamental equality -- |g 4.7. |t Another approach to the orbital integrals -- |g 4.8. |t locally symmetric space Z -- |g 5.1. |t operator Pa, y0 n and the function Jγ (Y0) -- |g 5.2. |t conjugate operator -- |g 5.3. |t evaluation of certain infinite dimensional traces -- |g 5.4. |t Some formulas of linear algebra -- |g 5.5. |t formula for Jγ(Y0) -- |g 6.1. |t Orbital integrals for the heat kernel -- |g 6.2. |t formula for general orbital integrals -- |g 6.3. |t orbital integrals for the wave operator -- |g 7.1. |t Characteristic forms on X -- |g 7.2. |t vector bundle of spinors on X and the Dirac operator -- |g 7.3. |t McKean-Singer formula on Z -- |g 7.4. |t Orbital integrals and the index theorem -- |g 7.5. |t proof of (7.4.4) -- |g 7.6. |t case of complex symmetric spaces -- |g 7.7. |t case of an elliptic element -- |g 7.8. |t de Rham-Hodge operator -- |g 7.9. |t integrand of de Rham torsion -- |g 8.1. |t case where G = K -- |g 8.2. |t case a not = to 0 [(γ), po] = 0 -- |g 8.3. |t case where G = SL2 (R) -- |g 9.1. |t Estimates on the heat kernel qxb, t away from iaN(k-1 -- |g 9.2. |t rescaling on the coordinates (f, Y) -- |g 9.3. |t conjugation of the Clifford variables -- |g 9.4. |t norm of α -- |g 9.5. |t conjugation of the hypoelliptic Laplacian -- |g 9.6. |t limit of the rescaled heat kernel -- |g 9.7. |t proof of Theorem 6.1.1 -- |g 9.8. |t translation on the variable YTX -- |g 9.9. |t coordinate system and a trivialization of the vector bundles -- |g 9.10. |t asymptotics of the operator pXA, a, B, YT0 AS B -> + infinity -- |g 9.11. |t proof of Theorem 9.6.1 -- |g 10.1. |t variational problem -- |g 10.2. |t Pontryagin maximum principle -- |g 10.3. |t variational problem on an Euclidean vector space -- |g 10.4. |t Mehler's formula -- |g 10.5. |t hypoelliptic heat kernel on an Euclidean vector space -- |g 10.6. |t Orbital integrals on an Euclidean vector space -- |g 10.7. |t Some computations involving Mehler's formula -- |g 10.8. |t probabilistic interpretation of the harmonic oscillator -- |g 11.1. |t scalar operators Axb, Bxb on X -- |g 11.2. |t Littlewood-Paley decomposition along the fibres TX -- |g 11.3. |t Littlewood-Paley decomposition on X -- |g 11.4. |t Littlewood Paley decomposition on X -- |g 11.5. |t heat kernels for Axb, Bxb -- |g 11.6. |t scalar hypoelliptic operators on X -- |g 11.7. |t scalar hypoelliptic operator on X with a quartic term -- |g 11.8. |t heat kernel associated with the operator LxA, b -- |g 12.1. |t Malliavin calculus for the Brownian motion on X -- |g 12.2. |t probabilistic construction of exp ( -tBxb)) over X -- |g 12.3. |t operator 136 and the wave equation -- |g 12.4. |t Malliavin calculus for the operator Bic, -- |g 12.5. |t tangent variational problem and integration by parts -- |g 12.6. |t uniform control of the integration by parts formula as b-> 0 -- |g 12.7. |t Uniform rough estimates on rxb, t, for bounded b -- |g 12.8. |t limit as b -> 0 -- |g 12.9. |t rough estimates as b -> + infinity -- |g 12.10. |t heat kernel rxb, t on X -- |g 12.11. |t heat kernel rxb, t on X -- |g 13.1. |t Hessian of the distance function -- |g 13.2. |t Bounds on the scalar heat kernel on X for bounded b -- |g 13.3. |t Bounds on the scalar heat kernel on X for bounded b -- |g 14.1. |t probabilistic construction of exp ( -tLxA)) -- |g 14.2. |t operator Lxb and the wave equation -- |g 14.3. |t Changing Y into -Y -- |g 14.4. |t probabilistic construction of exp ( -tLx'A, b) -- |g 14.5. |t Estimating V -- |g 14.6. |t Estimating W -- |g 14.7. |t proof of (4.5.3) when E is trivial -- |g 14.8. |t proof of the estimate (4.5.3) in the general case -- |g 14.9. |t Rough estimates on the derivatives of qx'b, t for bounded b -- |g 14.10. |t behavior of V as b -> 0 -- |g 14.11. |t limit of qx'b, t as b -> 0 -- |g 15.1. |t Uniform estimates on the kernel rxb, t over X -- |g 15.2. |t deviation from the geodesic flow for large b -- |g 15.3. |t scalar heat kernel on X away from Fγ = iaX(γ) -- |g 15.4. |t Gaussian estimates for rxb near iaX(γ) -- |g 15.5. |t scalar heat kernel on X away from Fγ=iaN(k-1) -- |g 15.6. |t Estimates on the scalar heat kernel on X near iaN(k-1) -- |g 15.7. |t proof of Theorem 9.1.1 -- |g 15.8. |t proof of Theorem 9.1.3 -- |g 15.9. |t proof of Theorem 9.5.6 -- |g 15.10. |t proof of Theorem 9.11.1. |
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