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Universal bi-Hamiltonian hierarchies of group-invariant (multicomponent) soliton equations are derived from non-stretching geometric curve flows $\map(t,x)$ in Riemannian symmetric spaces $M=G/H$, including compact semisimple Lie groups…
Let G be a connected complex simple Lie group with maximal compact subgroup U. Let g be the Lie algebra of G, and X = G/U be the associated Riemannian globally symmetric space of type IV. We have constructed three types of arithmetic…
The covariogram g_K of a convex body K in E^d is the function which associates to each x in E^d the volume of the intersection of K with K+x. In 1986 G. Matheron conjectured that for d=2 the covariogram g_K determines K within the class of…
We discuss global tensor invariants of a rigid body motion in the cases of Euler, Lagrange and Kovalevskaya. These invariants are obtained by substituting tensor fields with cubic on variable components into the invariance equation and…
Valuations on the space of finite-valued convex functions on $\mathbb{C}^n$ that are continuous, dually epi-translation invariant, as well as $\mathrm{U}(n)$-invariant are completely classified. It is shown that the space of these…
Recently, Bo'az Klartag showed that arbitrary convex bodies have Gaussian marginals in most directions. We show that Klartag's quantitative estimates may be improved for many uniformly convex bodies. These include uniformly convex bodies…
Complexifying space time has many interesting applications, from the construction of higher dimensional unification, to provide a useful framework for quantum gravity and to better define some local symmetries that suffer singularities in…
In this note, we give an explicit formula for a family of deformation quantizations for the momentum map associated with the cotangent lift of a Lie group action on Rd. This family of quantizations is parametrized by the formal G-systems…
We introduce a new operation between nonnegative integrable functions on $\mathbb{R} ^n$, that we call geometric combination; it is obtained via a mass transportation approach, playing with inverse distribution functions. The main feature…
Let G be a p-adic reductive group, and R an algebraically closed field. Let us consider a smooth representation of G on an R-vector space V. Fix an open compact subgroup K of G and a smooth irreducible representation of K on a…
We first introduce an invariant index for G-equivariant elliptic differential operators on a locally compact manifold M admitting a proper cocompact action of a locally compact group G. It generalizes the Kawasaki index for orbifolds to the…
The possible role of gravity in a noncommutative geometry is investigated. Due to the Moyal *-product of fields in noncommutative geometry, it is necessary to complexify the metric tensor of gravity. We first consider the possibility of a…
We obtain new invariant Einstein metrics on the compact Lie group $\SU(N)$ which are not naturally reductive. This is achieved by using the generalized flag manifold $G/K=\SU(k_1+\cdots +k_p)/\s(\U(k_1)\times\cdots\times\U(k_p))$ and by…
We consider the action of a noncompact torus H on the compact quotient G/L, where G is a Lie group containing H and L is a uniform lattice in G. Using harmonic analysis on G we prove a formula relating the compact orbits of H to the action…
To any graph with external half-edges and internal masses, we associate canonical integrals which depend non-trivially on particle masses and momenta, and are always finite. They are generalised Feynman integrals which satisfy graphical…
We propose a new method for obtaining Poincare-type inequalities on arbitrary convex bodies in R^n. Our technique involves a dual version of Bochner's formula and a certain moment map, and it also applies to some non-convex sets. In…
We study a compact invariant convex set $E$ in a polar representation of a compact Lie group. Polar rapresentations are given by the adjoint action of $K$ on $\mathfrak{p}$, where $K$ is a maximal compact subgroup of a real semisimple Lie…
The aim of this paper is to study properties of sections of convex bodies with respect to different types of measures. We present a formula connecting the Minkowski functional of a convex symmetric body K with the measure of its sections.…
Let G/K be a Riemannian symmetric space of the complex type, meaning that G is complex semisimple and K is a compact real form. Now let {\Gamma} be a discrete subgroup of G that acts freely and cocompactly on G/K. We consider the…
A generalization of the notion of a (pseudo-) Riemannian space is proposed in a framework of noncommutative geometry. In particular, there are parametrized families of generalized Riemannian spaces which are deformations of classical…