Papers by Valentin Ovsienko
The space of symbols of differential operators on a smooth manifold (i.e., the space of symmetric... more The space of symbols of differential operators on a smooth manifold (i.e., the space of symmetric contravariant tensor fields) is naturally a module over the Lie algebra of vector fields. We study, in the case of $\bf R^n$ with $n\geq2$, multi-parameter formal deformations of this module. The space of linear differential operators on $\bf R^n$ provides an important class of
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The first group of differentiable cohomology of $\Diff(S^1)$, vanishing on the M\"obius subg... more The first group of differentiable cohomology of $\Diff(S^1)$, vanishing on the M\"obius subgroup $PSL(2,R)\subset\Diff(S^1)$, with coefficients in modules of linear differential operators on $S^1$ is calculated. We introduce three non-trivial $PSL(2,R)$-invariant 1-cocycles on $\Diff(S^1)$ generalizing the Schwarzian derivative.
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The spaces of linear differential operators on ${\mathbb{R}}^n$ acting on tensor densities of deg... more The spaces of linear differential operators on ${\mathbb{R}}^n$ acting on tensor densities of degree $\lambda$ and the space of functions on $T^*{\mathbb{R}}^n$ which are polynomial on the fibers are not isomorphic as modules over the Lie algebra $\Vect({\mathbb{R}}^n)$ of vector fields on ${\mathbb{R}}^n$. However, these modules are isomorphic as $sl(n+1,{\mathbb{R}})$-modules where $sl(n+1,{\mathbb{R}})\subset \Vect({\mathbb{R}}^n)$ is the Lie algebra of infinitesimal projective
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Let ${\cal D}^k$ be the space of $k$-th order linear differential operators on ${\bf R}$: $A=a_k(... more Let ${\cal D}^k$ be the space of $k$-th order linear differential operators on ${\bf R}$: $A=a_k(x)\frac{d^k}{dx^k}+\cdots+a_0(x)$. We study a natural 1-parameter family of $\Diff(\bf R)$- (and $\Vect(\bf R)$)-modules on ${\cal D}^k$. (To define this family, one considers arguments of differential operators as tensor-densities of degree $\lambda$.) In this paper we solve the problem of isomorphism between $\Diff(\bf R)$-module structures on
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Comptes Rendus de l Académie des Sciences - Series I - Mathematics
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We discuss the theorem on the existence of six points on a convex closed plane curve in which the... more We discuss the theorem on the existence of six points on a convex closed plane curve in which the curve has a contact of order six with the osculating conic. (This is the ``projective version'' of the well known four vertices theorem for a curve in the Euclidean plane.) We obtain this classical fact as a corollary of some general Sturm-type theorems.
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We define the unique (up to normalization) symbol map from the space of linear differential opera... more We define the unique (up to normalization) symbol map from the space of linear differential operators on $R^n$ to the space of polynomial on fibers functions on $T^* R^n$, equivariant with respect to the Lie algebra of projective transformations $sl_{n+1}\subset\Vect(R^n)$. We apply the constructed $sl_{n+1}$-invariant symbol to studying of the natural one-parameter family of $\Vect(M)$-modules on the space of linear differential operators on an arbitrary manifold M. Each of the $\Vect(M)$-action from this family can be interpreted as a deformation of the standard $\Vect(M)$-module $S(M)$ of symmetric contravariant tensor fields on M. We define (and calculatein the case: $M= R^n$) the corresponding cohomology of $\Vect(M)$ related with this deformation. This cohomology realize the obstruction for existence of equivariant symbol and quantization maps. The projective Lie algebra $sl_{n+1}$ naturally appears as the algebra of symmetries on which the involved $\Vect(M)$-cohomology is tr...
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Let ${\cal F}_\lambda$ be the space of tensor densities on ${\bf R}^n$ of degree $\lambda$ (or, e... more Let ${\cal F}_\lambda$ be the space of tensor densities on ${\bf R}^n$ of degree $\lambda$ (or, equivalently, of conformal densities of degree $-\lambda{}n$) considered as a module over the Lie algebra $so(p+1,q+1)$. We classify $so(p+1,q+1)$-invariant bilinear differential operators from ${\cal F}_\lambda\otimes{\cal F}_\mu$ to~${\cal F}_\nu$. The classification of linear $so(p+1,q+1)$-invariant differential operators from ${\cal F}_\lambda$ to ${\cal F}_\mu$ already known in the literature is obtained in a different manner.
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Notices of the American Mathematical Society
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The pentagram map is a projectively natural iteration defined on polygons, and also on objects we... more The pentagram map is a projectively natural iteration defined on polygons, and also on objects we call twisted polygons. (A twisted polygon is a map from Z into the projective plane that is periodic modulo a projective transformation.) We find a Poisson structure on the space of twisted polygons and show that the pentagram map relative to this Poisson structure is completely integrable in the sense of Arnold-Liouville. For certain families of twisted polygons, such as those we call universally convex, we translate the integrability into a statement about the quasi-periodic motion for the dynamics of the pentagram map. We also explain how the pentagram map, in the continuous limit, corresponds to the classical Boussinesq equation. The Poisson structure we attach to the pentagram map is a discrete version of the first Poisson structure associated with the Boussinesq equation.
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We consider a family, $\F$, of subsets of an $n$-set such that the cardinality of the symmetric d... more We consider a family, $\F$, of subsets of an $n$-set such that the cardinality of the symmetric difference of any two elements $F,F'\in\F$ is not a multiple of 4. We prove that the maximal size of $\F$ is bounded by $2n$, unless $n\equiv{}3\mod4$ when it is bounded by $2n+2$. Our method uses cubic forms on $\bbF_2^n$ and the Hurwitz-Radon theory of square identities. We also apply this theory to obtain some information about boolean cubic forms and so-called additive quadruples.
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We consider differential operators between sections of arbitrary powers of the determinant line b... more We consider differential operators between sections of arbitrary powers of the determinant line bundle over a contact manifold. We extend the standard notions of the Heisenberg calculus: noncommutative symbolic calculus, the principal symbol, and the contact order to such differential operators. Our first main result is an intrinsically defined "subsymbol" of a differential operator, which is a differential invariant of degree one lower than that of the principal symbol. In particular, this subsymbol associates a contact vector field to an arbitrary second order linear differential operator. Our second main result is the construction of a filtration that strengthens the well-known contact order filtration of the Heisenberg calculus.
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We consider the universal central extension of the Lie algebra Vect(S 1 )nC 1 (S 1 ). The coadjoi... more We consider the universal central extension of the Lie algebra Vect(S 1 )nC 1 (S 1 ). The coadjoint representation of this Lie algebra has a natural geometric interpretation by matrix analogues of the Sturm-Liouville operators. This approach leads to new Lie superalgebras generalizing the well-known Neveu-Schwartz algebra. 1 Introduction 1.1 Sturm-Liouville operators and the action of Vect(S 1 ). Let us recall some well-known definitions (cf. e.g. [9],[8]). Consider the Sturm-Liouville operator: L = Gamma2c d 2 dx 2 + u(x) (1) where c 2 R and u is a periodic potential: u(x + 2ß) = u(x) 2 C 1 (R). Let Vect(S 1 ) be the Lie algebra of smooth vector field on S 1 : f = f(x) d dx ; where f(x + 2ß) = f(x), with the commutator [f(x) d dx ; g(x) d dx ] = (f(x)g 0 (x) Gamma f 0 (x)g(x)) d dx : We define a Vect(S 1 )-action on the space of Sturm-Liouville operators. Consider a 1-parameter family of Vect(S 1 )-actions on the space of smooth functions C 1 (S 1...
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The Arnold-Gelfand Mathematical Seminars, 1996
ABSTRACT We discuss the theorem on the existence of six points on a convex closed plane curve in ... more ABSTRACT We discuss the theorem on the existence of six points on a convex closed plane curve in which the curve has a contact of order six with the osculating conic. (This is the ``projective version'' of the well known four vertices theorem for a curve in the Euclidean plane.) We obtain this classical fact as a corollary of some general Sturm-type theorems.
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Let $(M,g)$ be a pseudo-Riemannian manifold and $F_\lambda(M)$ the space of densities of degree $... more Let $(M,g)$ be a pseudo-Riemannian manifold and $F_\lambda(M)$ the space of densities of degree $\lambda$ on $M$. We study the space $D^2_{\lambda,\mu}(M)$ of second-order differential operators from $F_\lambda(M)$ to $F_\mu(M)$. If $(M,g)$ is conformally flat with signature $p-q$, then $D^2_{\lambda,\mu}(M)$ is viewed as a module over the group of conformal transformations of $M$. We prove that, for almost all values
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Noncommutative Differential Geometry and Its Applications to Physics, 2001
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Papers by Valentin Ovsienko