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Semyon Alesker

Publications and source records attributed to Semyon Alesker.

At least 37 records · Page 2Linked to original sources
arXiv2015

Convolution of valuations on manifolds

We introduce the new notion of convolution of a (smooth or generalized) valuation on a group $G$ and a valuation on a manifold $M$ acted upon by the group. In the case of a transitive group action, we prove that the spaces of smooth and generalized valuations on $M$ are modules over the algebra of compactly supported generalized valuations on $G$ satisfying some technical condition of tameness. The case of a vector space acting on itself is studied in detail. We prove explicit formulas in this case and show that the new convolution is an extension of the convolution on smooth translation invariant valuations introduced by J.~Fu and the second named author.

math.DG↗
arXiv2015

On repeated sequential closures of constructible functions in valuations

The space of constructible functions form a dense subspace of the space of generalized valuations. In this note we prove a somewhat stronger property that the sequential closure, taken sufficiently many (in fact, infinitely many) times, of the former space is equal to the latter one. This stronger property is necessary for some applications in the theory of valuations on manifolds.

math.MG↗
arXiv2015

On an analytic description of the $α$-cosine transform on real Grassmannians

The goal of this paper is to describe the $α$-cosine transform on functions on a Grassmannian of $i$-planes in an $n$-dimensional real vector space. in analytic terms as explicitly as possible. We show that for all but finitely many complex $α$ the $α$-cosine transform is a composition of the $(α+2)$-cosine transform with an explicitly written (though complicated) O(n)-invariant differential operator. For all exceptional values of $α$ except one we interpret the $α$-cosine transform explicitly as either the Radon transform or composition of two Radon transforms. Explicit interpretation of the transform corresponding to the last remaining value $α$, which is $-(min\{i,n-i\}+1)$, is still an open problem.

math.MG↗
arXiv2014

Valuations on manifolds and integral geometry

One constructs new operations of pull-back and push-forward on valuations on manifolds with respect to submersions and immersions. A general Radon type transform on valuations is introduced using these operations and the product on valuations. It is shown that the classical Radon transform on smooth functions, and the well known Radon transform on constructible functions with respect to the Euler characteristic are special cases of this new Radon transform. An inversion formula for the Radon transform on valuations has been proven in a specific case of real projective spaces. Relations of these operations to yet another classical type of integral geometry, Crofton and kinematic formulas, are indicated.

math.MG↗
arXiv2014

Cohomology of a Hamiltonian T-space with involution

Let $M$ be a compact symplectic manifold on which a compact torus $T$ acts Hamiltonialy with a moment map $μ$. Suppose there exists a symplectic involution $θ:M\to M$, such that $μ\circθ=-μ$. Assuming that 0 is a regular value of $μ$, we calculate the trace of the action of $θ$ on the cohomology of $M$ in terms of the trace of the action of $θ$ on the symplectic reduction $μ^{-1}(0)/T$ of $M$. This result generalizes a theorem of R. Stanley, who considered the case when $M$ was a toric variety.

math.SG↗
arXiv2013

On extendability by continuity of valuations on convex polytopes

There is a well known construction of weakly continuous valuations on convex compact polytopes in R^n. In this paper we investigate when a special case of this construction gives a valuation which extends by continuity in the Hausdorff metric to all convex compact subsets of R^n. It is shown that there is a necessary condition on the initial data for such an extension. In the case of R^3 more explicit results are obtained.

math.MG↗
arXiv2013

The product on smooth and generalized valuations

The product of smooth valuations on manifolds is described in terms of differential forms, Gelfand transforms and blow-up spaces. It is shown that the product extends partially to generalized valuations and corresponds geometrically to transversal intersections. This result is used to prove a general kinematic formula on compact rank one symmetric spaces.

math.MG↗
arXiv2013

A Fourier type transform on translation invariant valuations on convex sets

Let $V$ be a finite dimensional real vector space. In the article we construct an isomorphism between the space of smooth translation invariant valuations on convex subsets of $V$ and the space of such valuations (twisted by densities) on the dual space $V^*$. This isomorphism commutes with the natural action of the full linear group on both spaces of valuations, maps product on the source (introduced in arXiv:math/0301148) to the convolution on the target (introduced in arXiv:math/0607449), and satisfies a Plancherel type inversion formula. As an application, a version of the hard Lefschetz theorem for valuations is proved.

math.MG↗
arXiv2011

Pluripotential theory on quaternionic manifolds

On any quaternionic manifold of dimension greater than 4 a class of plurisubharmonic functions (or, rather, sections of an appropriate line bundle) is introduced. Then a Monge-Ampère operator is defined. It is shown that it satisfies a version of theorems of A. D. Alexandrov and Chern-Levine-Nirenberg. These notions and results were previously known in the special case of hypercomplex manifolds. One of the new technical aspects of the present paper is the systematic use of the Baston differential operators, for which we prove a new multiplicativity property.

math.CV↗
arXiv2011

On modules over valuations

Recently an algebra of smooth valuations was attached to any smooth manifold. Roughly put, a smooth valuation is finitely additive measure on compact submanifolds with corners which satisfies some extra properties. In this note we initiate a study of modules over smooth valuations. More specifically we study finitely generated projective modules in analogy to the study of vector bundles on a manifold. In particular it is shown that on a compact manifold there exists a canonical isomorphism between the $K$-ring constructed out of finitely generated projective modules over valuations and the classical topological $K^0$-ring constructed out of vector bundles.

math.MG↗
arXiv2011

Harmonic analysis of translation invariant valuations

The decomposition of the space of continuous and translation invariant valuations into a sum of SO(n) irreducible subspaces is obtained. A reformulation of this result in terms of a Hadwiger type theorem for continuous translation invariant and SO(n)-equivariant tensor valuations is also given. As an application, symmetry properties of rigid motion invariant and homogeneous bivaluations are established and then used to prove new inequalities of Brunn-Minkowski type for convex body valued valuations.

math.DG↗
arXiv2010

New structures on valuations and applications

An overview of some of the recent developments in the theory of valuations on convex sets and its generalizations to manifolds is given. The exposition is focused towards applications to integral geometry; several of such applications are discussed.

math.MG↗
arXiv2008

Quaternionic Monge-Ampere equation and Calabi problem for HKT-manifolds

A quaternionic version of the Calabi problem on Monge-Ampere equation is introduced. It is a quaternionic Monge-Ampere equation on a compact hypercomplex manifold with an HKT-metric. The equation is non-linear elliptic of second order. For a hypercomplex manifold with holonomy in SL(n;H), uniqueness (up to a constant) of a solution is proven, as well as the zero order a priori estimate. The existence of solution is conjectured, similar to Calabi-Yau theorem. We reformulate this quaternionic equation as a special case of a complex Hessian equation, making sense on any complex manifold.

math.CV↗
arXiv2006

Theory of valuations on manifolds, IV. New properties of the multiplicative structure

This is the fourth part in the series of articles math.MG/0503397, math.MG/0503399, math.MG/0509512 where the theory of valuations on manifolds is developed. In this part it is shown that the filtration on valuations introduced in math.MG/0503399 is compatible with the product. Then it is proved that the Euler-Verdier involution on smooth valuations introduced in math.MG/0503399 is an automorphism of the algebra of valuations. Then an integration functional on valuations with compact support is introduced, and a property of selfduality of valuations is proved. Next a space of generalized valuations is defined, and some basic properties of it are proved. Finally a canonical imbedding of the space of constructible functions on a real analytic manifold into the space of generalized valuations is constructed, and various structures on valuations are compared with known structures on constructible functions.

math.MG↗
arXiv2006

Theory of valuations on manifolds: a survey

This is a non-technical survey of a recent theory of valuations on manifolds constructed in math.MG/0503397, math.MG/0503399, math.MG/0509512, math.MG/0511171 and actually a guide to this series of articles. We review also some recent related results obtained by a number of people. We formulate some open questions.

math.MG↗
arXiv2005

Theory of valuations on manifolds, III. Multiplicative structure in the general case

This article is the third part of the series of articles where the theory of valuations on manifolds is constructed. In math.MG/0503399 the notion of a smooth valuation on a manifold was introduced. The goal of this article is to put a canonical multiplicative structure on the space of smooth valuations on general manifolds, thus extending some of the affine constructions from math.MG/0301148, math.MG/0503397.

math.MG↗
arXiv2005

Theory of valuations on manifolds, II

This article is the second part in the series of articles where we are developing theory of valuations on manifolds. Roughly speaking valuations could be thought as finitely additive measures on a class of nice subsets of a manifold which satisfy some additional assumptions. The goal of this article is to introduce a notion of a smooth valuation on an arbitrary smooth manifold and establish some of the basic properties of it.

math.MG↗