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53
Conformally equivariant quantization: Existence and uniqueness
"... We prove the existence and the uniqueness of a conformally equivariant symbol calculus and quantization on any conformally flat pseudoRiemannian manifold (M,g). In other words, we establish a canonical isomorphism between the spaces of polynomials on T ∗ M and of differential operators on tensor de ..."
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Cited by 41 (5 self)
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We prove the existence and the uniqueness of a conformally equivariant symbol calculus and quantization on any conformally flat pseudoRiemannian manifold (M,g). In other words, we establish a canonical isomorphism between the spaces of polynomials on T ∗ M and of differential operators on tensor densities over M, both viewed as modules over the Lie algebra o(p + 1,q + 1) where p + q = dim(M). This quantization exists for generic values of the weights of the tensor densities and compute the critical values of the weights yielding obstructions to the existence of such an isomorphism. In the particular case of halfdensities, we obtain a conformally invariant starproduct.
Appendix  Projective Geometry for Machine Vision
, 1992
"... Introduction The idea for this Appendix arose from our perception of a frustrating situation faced by vision researchers. For example, one is interested in some aspect of the theory of perspective image formation such as the epipolar line. The interested party goes to the library to check out a boo ..."
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Cited by 27 (0 self)
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Introduction The idea for this Appendix arose from our perception of a frustrating situation faced by vision researchers. For example, one is interested in some aspect of the theory of perspective image formation such as the epipolar line. The interested party goes to the library to check out a book on projective geometry filled with hope that the necessary mathematical machinery will be directly at hand. These expectations are quickly dashed. Upon opening the book, the expectant reader finds the presentation dominated by endless observations about harmonic relations and a few chapters which explore the minutiae of Pappus' theorem. Finally, as a last cruel twist of irony, the book ends in triumph with a rather exhilarating discourse on the conic pencil. All of the material is presented in the form of theorems defined on points, lines and conics without the use of coordinates, except perhaps for a quick pause to define barycentric coordinates just to taunt the reader. Dejected, the vis
Projectively equivariant symbol calculus
, 1999
"... The spaces of linear differential operators Dλ(R n) acting on λdensities on R n and the space Pol(T ∗ R n) of functions on T ∗ R n which are polynomial on the fibers are not isomorphic as modules over the Lie algebra Vect(R n) of vector fields of R n. However, these modules are isomorphic as sl(n + ..."
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Cited by 26 (2 self)
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The spaces of linear differential operators Dλ(R n) acting on λdensities on R n and the space Pol(T ∗ R n) of functions on T ∗ R n which are polynomial on the fibers are not isomorphic as modules over the Lie algebra Vect(R n) of vector fields of R n. However, these modules are isomorphic as sl(n + 1, R)modules where sl(n + 1, R) ⊂ Vect(R n) is the Lie algebra of infinitesimal projective transformations. In addition, such an sln+1equivariant bijection is unique (up to normalization). This leads to a notion of projectively equivariant quantization and symbol calculus for a manifold endowed with a (flat) projective structure. We apply the sln+1equivariant symbol map to study the Vect(M)modules D k λ (M) of kthorder linear differential operators acting on λdensities, for an arbitrary manifold M and classify the quotientmodules D k λ (M)/Dℓ λ (M). 1
Normalizability of Onedimensional Quasiexactly Solvable Schrödinger Operators
"... We completely determine necessary and sufficient conditions for the normalizability of the wave functions giving the algebraic part of the spectrum of a quasiexactly solvable Schrodinger operator on the line. Methods from classical invariant theory are employed to provide a complete list of canonica ..."
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Cited by 20 (11 self)
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We completely determine necessary and sufficient conditions for the normalizability of the wave functions giving the algebraic part of the spectrum of a quasiexactly solvable Schrodinger operator on the line. Methods from classical invariant theory are employed to provide a complete list of canonical forms for normalizable quasiexactly solvable Hamiltonians and explicit normalizability conditions in general coordinate systems.
Space of linear differential operators on the real line as a module over the Lie algebra of vector fields
, 1996
"... Let Dk be the space of kth order linear differential operators on R: A = ak(x) dk dxk + · · · + a0(x). We study a natural 1parameter family of Diff(R) (and Vect(R))modules on Dk. (To define this family, one considers arguments of differential operators as tensordensities of degree λ.) In this ..."
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Cited by 14 (8 self)
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Let Dk be the space of kth order linear differential operators on R: A = ak(x) dk dxk + · · · + a0(x). We study a natural 1parameter family of Diff(R) (and Vect(R))modules on Dk. (To define this family, one considers arguments of differential operators as tensordensities of degree λ.) In this paper we solve the problem of isomorphism between Diff(R)module structures on Dk corresponding to different values of λ. The result is as follows: for k = 3 Diff(R)module structures on D3 are isomorphic to each other for every values of λ ̸ = 0, 1, 1 2
Characterization of Hermitian symmetric spaces by fundamental forms
 621–634. LIE ALGEBRA COHOMOLOGY, AND RIGIDITY 27
"... We show that an equivariantly embedded Hermitian symmetric space in a projective space which contains neither a projective space nor a hyperquadric as a component is characterized by its fundamental forms as a local submanifold of the projective space. Using some invarianttheoretic properties of th ..."
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Cited by 10 (0 self)
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We show that an equivariantly embedded Hermitian symmetric space in a projective space which contains neither a projective space nor a hyperquadric as a component is characterized by its fundamental forms as a local submanifold of the projective space. Using some invarianttheoretic properties of the fundamental forms and Seashi’s work on linear differential equations of finite type, we reduce the proof to the vanishing of certain Spencer cohomology groups. The vanishing is checked by Kostant’s harmonic theory for Lie algebra cohomology.
On The Rigidity Of Differential Systems Modelled On Hermitian Symmetric Spaces And Disproofs Of A Conjecture Concerning Modular Interpretations Of Configuration Spaces
 ADV. STUDIES IN PURE MATH
, 1994
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Conformally equivariant quantization
"... Let (M,g) be a pseudoRiemannian manifold and Fλ(M) the space of densities of degree λ on M. We study the space D2 λ,µ (M) of secondorder differential operators from Fλ(M) to Fµ(M). If (M,g) is conformally flat with signature p −q, then D2 λ,µ (M) is viewed as a module over the group of conformal t ..."
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Cited by 9 (1 self)
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Let (M,g) be a pseudoRiemannian manifold and Fλ(M) the space of densities of degree λ on M. We study the space D2 λ,µ (M) of secondorder differential operators from Fλ(M) to Fµ(M). If (M,g) is conformally flat with signature p −q, then D2 λ,µ (M) is viewed as a module over the group of conformal transformations of M. We prove that, for almost all values of µ − λ, the O(p+1,q+1)modules D2 λ,µ (M) and the space of symbols (i.e., of secondorder polynomials on T ∗M) are canonically isomorphic. This yields a conformally equivariant quantization for quadratic Hamiltonians. We furthermore show that this quantization map extends to arbitrary pseudoRiemannian manifolds and depends only on the conformal class [g] of the metric. As an example, the quantization of the geodesic flow yields a novel conformally equivariant Laplace operator on halfdensities, as well as the wellknown Yamabe Laplacian. We also recover in this framework the multidimensional Schwarzian derivative of conformal transformations.
Generalized Wilczynski invariants for nonlinear ordinary differential equations
, 2007
"... Abstract. We show that classical Wilczynski–Seashi invariants of linear systems of ordinary differential equations are generalized in a natural way to contact invariants of nonlinear ODEs. We explore geometric structures associated with equations that have vanishing generalized Wilczynski invarian ..."
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Cited by 8 (2 self)
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Abstract. We show that classical Wilczynski–Seashi invariants of linear systems of ordinary differential equations are generalized in a natural way to contact invariants of nonlinear ODEs. We explore geometric structures associated with equations that have vanishing generalized Wilczynski invariants and establish relationship of such equations with deformation theory of rational curves on complex algebraic surfaces. 1.