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In the paper a Riemannian structure on the tangent bundle is defined by using a statistical structure $(g,\nabla)$ on the base manifold. Expressions for various curvatures of the structure are derived. Some rigidity results of the structure…
We study multivariate polynomials over `structured' grids. We begin by proposing an interpretation as to what it means for a finite subset of a field to be structured; we do so by means of a numerical parameter, the nullity. We then extend…
Two new classes of metrizable vector bundles have been presented in the papers [1] and [4]. The Lie algebroid generalized tangent bundle of a dual vector bundle is presented. This Lie algebroid is a new example of metrizable vector bundle.…
A notion of an algebroid - a generalization of a Lie algebroid structure is introduced. We show that many objects of the differential calculus on a manifold M associated with the canonical Lie algebroid structure on T^M can be obtained in…
This article gives a survey of recent results on a generalization of the notion of a Hodge structure. The main example is related to the Fourier-Laplace transform of a variation of polarizable Hodge structure on the punctured affine line,…
Natural metric structures on tangent bundles and tangent sphere bundles enclose many important problems, from the topology of the base to the determination of their holonomy. We make here a brief study of the topic. We find the…
Motivated by the computations done in \cite{C1}, where I introduced and discussed what I called the groupoid of generalized gauge transformations, viewed as a groupoid over the objects of the category $\mathsf{Bun}_{G,M}$ of principal…
This is a report on recent progress concerning the interactions between derived algebraic geometry and deformation quantization. We present the notion of derived algebraic stacks, of shifted symplectic and Poisson structures, as well as the…
An operator generalisation of the notion of geometric phase has been recently proposed purely based on physical grounds. Here we provide a mathematical foundation for its existence, while uncovering new geometrical structures in quantum…
We consider plumbings of symplectic disk bundles over spheres admitting concave contact boundary, with the goal of understanding the geometric properties of the boundary contact structure in terms of the data of the plumbing. We focus on…
In this paper we address what generalized geometric structures are possible on products of spaces that each admit generalized geometries. In particular we consider, first, the product of two odd dimensional spaces that each admit a…
A class of metrizable vector bundles in the general framework of generalized Lie algebroids have been presented in the eight reference. Using a generalized Lie algebroid we obtain the Lie algebroid generalized tangent bundle of a vector…
We introduce a general theory of parametrized objects in the setting of infinity categories. Although spaces and spectra parametrized over spaces are the most familiar examples, we establish our theory in the generality of objects of a…
We consider a geometrization, i.e., we identify geometrical structures, for the space of density states of a quantum system. We also provide few comments on a possible application of this geometrization for composite systems.
Let f be a generic polynomial mapping mapping from the plane to the plane. There are constructed quadratic forms whose signatures determine the number of positive and negative cusps of f.
From a geometrical point of view it is, so far, not sufficiently well understood what should be a "noncommutative principal bundle". Still, there is a well-developed abstract algebraic approach using the theory of Hopf algebras. An…
We study the notion of geometric structures for toposes: This generalizes the notion of (X,G) manifolds. We give some applications to algebraic geometry
The paper presents an extension of the geometric quantization procedure to integrable, big-isotropic structures. We obtain a generalization of the cohomology integrality condition, we discuss geometric structures on the total space of the…
Bundle gerbes are simple examples of higher geometric structures that show their utility in dealing with topological subtleties of physical theories. I review a recent construction of torsion topological invariants for condensed matter…
Cosymplectic and normal almost contact structures are analogues of symplectic and complex structures that can be defined on 3-manifolds. Their existence imposes strong topological constraints. Generalized geometry offers a natural common…