Related papers: Integral geometry and Mizel's problem
Although this article can be read independently, it is a continuation of the introduction to integrable systems aspects of quantum cohomology given in part 1 (math.DG/0104274). In the same elementary style, i.e. assuming basic properties of…
We introduce the combinatorial Lyubeznik resolution of monomial ideals. We prove that this resolution is isomorphic to the usual Lyubezbnik resolution. As an application, we give a combinatorial method to determine if an ideal is a…
We derive solvability conditions and closed-form solution for the Weber type integral equation, related to the familiar Weber-Orr integral transforms and the old Weber-Titchmarsh problem (posed in Proc. Lond. Math. Soc. 22 (2) (1924),…
It is shown that, given any finite dimensional, split basic algebra $\Lambda = K\Gamma/I$ (where $\Gamma$ is a quiver and $I$ an admissible ideal in the path algebra $K \Gamma$), there is a finite list of affine algebraic varieties, the…
We prove that the integral of a certain Riesz-type kernel over $(n-1)$-rectifiable sets in $\mathbb{R}^n$ is constant, from which a formula for surface measure immediately follows. Geometric interpretations are given, and the solution to a…
The discrete functional $L_p$ Minkowski problem is posed and solved. As a consequence, the general affine P\'{o}lya-Szeg\"{o} principle and the general affine Sobolev inequalities are established.
Hilbert's fourth problem asks for the construction and the study of metrics on subsets of projective space for which the projective line segments are geodesics. Several solutions of the problem were given so far, depending on more precise…
Partial Isometries are important constructs that help give nontrivial solutions once a simple solution is known. We generalize this notion to Extended Partial Isometries and include operators which have right inverses but no left inverses…
A new series of integrable cases of the many-body problem in many-dimensional spaces is found. That series appears as a part of the larger series of integrable problems, which are in 1-1 correspondence with Krichever-Novikov algebras of…
A list of open problems on holomorphic symplectic, contact and Poisson manifolds.
We consider gauge fields associated with a semisimple Malcev algebra. We construct a gauge-invariant Lagrangian and found a solution of modified Yang-Mills equations in seven dimensions.
A complete classification and character formulas for finite-dimensional irreducible representations of the rational Cherednik algebra of type A is given. Less complete results for other types are obtained. Links to the geometry of affine…
A survey of some recent and important results which have to do with integrable equations and their relationship with the theory of surfaces is given. Some new results are also presented. The concept of the moving frame is examined, and it…
The fundamental ideas of the definition of solvable and semisimple Bol algebras are given and some related theorems
The $2n$ dimensional manifold with two mutually commutative operators of differentiation is introduced. Nontrivial multidimensional integrable systems connected with arbitrary graded (semisimple) algebras are constructed. The general…
We study the Dirichlet problem for functions whose graphs are spacelike hypersurfaces with prescribed curvature in the Minkowski space and we obtain some new interior second order estimates for admissible solutions to the corresponding…
In this paper, we generalise an interesting geometry problem from the 1995 edition of the International Mathematical Olympiad (IMO) using analytic geometry tools.
We find a solution of Sincov's inequality. Further, we prove that in the differentiable case we can interpret such solution as a differentiable manifold in the original sense of Lang. This allows to generalize the notion of atlas and…
This is a survey of some problems in geometric group theory which I find interesting. The problems are from different areas of group theory. Each section is devoted to problems in one area. It contains an introduction where I give some…
The purpose of this paper is to extend the explicit geometric evaluation of semisimple orbital integrals for smooth kernels for the Casimir operator obtained by the first author to the case of kernels for arbitrary elements in the center of…