Related papers: Points on quantum projectivations
We prove finite generation of the algebras of invariants for a class of linear actions of suitable non-reductive groups on projective and affine varieties, and give a geometric construction for their GIT quotients.
The issue of whether some manifestations of gravitation in the quantum domain, are indicative or not of a non-geometrical aspect in gravitation is discussed. We examine some examples that have been considered in this context, providing a…
I hesitated for a long time before giving shape to these notes, originally intended for preliminary reading by the attendees to the Summer School "New paths towards quantum gravity" (Holbaek Bay, Denmark, May 2008). At the end, I decide…
In this paper we introduce the semi-graded rings, which extend graded rings and skew PBW extensions. For this new type of non-commutative rings we will discuss some basic problems of non-commutative algebraic geometry. In particular, we…
We survey some aspects of the theory of noncommutative manifolds focusing on the noncommutative analogs of two-dimensional tori and low-dimensional spheres. We are particularly interested in those aspects of the theory that link the…
Nonlinear analysis has played a prominent role in the recent developments in geometry and topology. The study of the Yang-Mills equation and its cousins gave rise to the Donaldson invariants and more recently, the Seiberg-Witten invariants.…
We study a deformation of infinitesimal diffeomorphisms of a smooth manifold. The deformation is based on a general twist. This leads to a differential geometry on a noncommutative algebra of functions whose product is a star-product. The…
In Noncommutative Geometry, as in quantum theory, classically real variables are assumed to correspond to self-adjoint operators. We consider the relaxation of the requirement of self-adjointness to mere symmetry for operators $X_i$ which…
Following the guidelines of classical differential geometry the `building material' for the tensor calculus in non-commutative geometry is suggested. The algebraic account of moduli of vectors and covectors is carried out.
A survey on recent developments in (algebraic) integral geometry is given. The main focus lies on algebraic structures on the space of translation invariant valuations and applications in integral geometry.
We construct a family of canonical connections and surrounding basic theory for almost complex manifolds that are equipped with an affine connection. This framework provides a uniform approach to treating a range of geometries. In…
Motivated by the fundamental results of the geometric algebra we study quadrilateral lattices in projective spaces over division rings. After giving the noncommutative discrete Darboux equations we discuss differences and similarities with…
Combining geometric group theory techniques with geometric topology tools, we show how primitive cohomologies provide useful insights towards unifying the mathematical formulation of Gromov-Witten invariants. In particular, we emphasise the…
We present alternative postulates for Euclidean geometry whose merit is that they lead to a new class of invariants and associated geometries for real finite-dimensional unital associative algebras.
This PhD thesis aims at describing the applications of noncommutative geometry to particle physics and quantum field theory. It includes a brief survey of the basic principles and definitions of noncommutative geometry such as spectral…
Geometric torsions are torsions of acyclic complexes of vector spaces which consist of differentials of geometric quantities assigned to the elements of a manifold triangulation. We use geometric torsions to construct invariants for a…
In this paper we apply a recently proposed algebraic theory of integration to projective group algebras. These structures have received some attention in connection with the compactification of the $M$ theory on noncommutative tori. This…
We examine some recent developments in noncommutative geometry, including spin geometries on noncommutative tori and their quantization by the Shale-Stinespring procedure, as well as the emergence of Hopf algebras as a tool linking index…
Motivated by applications in computational anatomy, we consider a second-order problem in the calculus of variations on object manifolds that are acted upon by Lie groups of smooth invertible transformations. This problem leads to solution…
We consider noncommutative geometries obtained from a triangular Drinfeld twist and review the formulation of noncommutative gravity. A detailed study of the abelian twist geometry is presented, including the fundamental theorem of…