We prove the existence and the uniqueness of a conformally equivariant symbol calculus and quantization on any conformally flat pseudo-Riemannian 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 half-densities, we obtain a conformally invariant star-product.

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

The spaces of linear differential operators Dλ(R n) acting on λ-densities 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+1-equivariant 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+1-equivariant symbol map to study the Vect(M)-modules D k λ (M) of k-th-order linear differential operators acting on λ-densities, for an arbitrary manifold M and classify the quotient-modules D k λ (M)/Dℓ λ (M). 1

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 quasi-exactly solvable Hamiltonians and explicit normalizability conditions in general coordinate systems.

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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 invariant-theoretic 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.

Let (M,g) be a pseudo-Riemannian manifold and Fλ(M) the space of densities of degree λ on M. We study the space D2 λ,µ (M) of second-order 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 second-order 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 pseudo-Riemannian 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 half-densities, as well as the well-known Yamabe Laplacian. We also recover in this framework the multi-dimensional Schwarzian derivative of conformal transformations.

We completely characterize all nonlinear partial differential equations leaving a given finite-dimensional vector space of analytic functions invariant. Existence of an invariant subspace leads to a reduction of the associated dynamical partial differential equations to a system of ordinary differential equations, and provide a nonlinear counterpart to quasi-exactly solvable quantum Hamiltonians. These results rely on a useful extension of the classical Wronskian determinant condition for linear independence of functions. In addition, new approaches to the characterization of the annihilating differential operators for spaces of analytic functions are presented.

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Abstract. We show that classical Wilczynski–Se-ashi invariants of linear systems of ordinary differential equations are generalized in a natural way to contact invariants of non-linear 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.