A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
Study cohomology of quaternionic foliations and orbifolds.
Using very weak criteria for what may constitute a noncommutative geometry, I show that a pseudo-Riemannian manifold can only be smoothly deformed into noncommutative geometries if certain geometric obstructions vanish. These obstructions can be expressed as a system of partial differential equations relating the metri…
Develops Riemannian geometry for noncommutative super surfaces.
Our understanding of the notion of curvature in a noncommutative setting has progressed substantially in the past ten years. This new episode in noncommutative geometry started when a Gauss-Bonnet theorem was proved by Connes and Tretkoff for a curved noncommutative two torus. Ideas from spectral geometry and heat kern…
New framework for noncommutative Carrollian geometry using Lie-Rinehart pairs.
The paper explores noncommutative geometry of frame bundles using C*-algebras.
A general question behind this paper is to explore a good notion for intrinsic curvature in the framework of noncommutative geometry started by Alain Connes in the 80s. It has only recently begun (2014) to be comprehended via the intensive study of modular geometry on the noncommutative two tori. In this paper, we exte…
This is the introduction and bibliography for lecture notes of a course given at the Summer School on Noncommutative Geometry and Applications, sponsored by the European Mathematical Society, at Monsaraz and Lisboa, Portugal, September 1-10, 1997. In the published version, an epilogue of recent developments and many ne…
New bridge between diffeology and noncommutative geometry.
In this paper, metric reduction in generalized geometry is investigated. We show how the Bismut connections on the quotient manifold are obtained from those on the original manifold. The result facilitates the analysis of generalized Khler reduction, which motivates the concept of metric generalized principal…
We review basic notions and methods of noncommutative geometry and their applications to analysis and geometry on foliated manifolds.
This paper defines and examines the basic properties of noncommutative analogues of almost complex structures, integrable almost complex structures, holomorphic curvature, cohomology, and holomorphic sheaves. The starting point is a differential structure on a noncommutative algebra defined in terms of a differential g…
We introduce a class of almost homogeneous varieties contained in the class of spherical varieties and containing horospherical varieties as well as complete symmetric varieties. We develop K{ä}hler geometry on these varieties, with applications to canonical metrics in mind, as a generalization of the Guillemin-Abreu-D…
Sheaves on graphs link to noncommutative geometry.
Introduces noncommutative geometry for modeling quantum spacetime.
Constructs noncommutative spaces for D-branes on complex algebraic spaces.
In this review we present some of the fundamental mathematical structures which permit to define noncommutative gauge field theories. In particular, we emphasize the theory of noncommutative connections, with the notions of curvatures and gauge transformations. Two different approaches to noncommutative geometry are co…
We review some applications of noncommutative geometry to the study of transverse geometry of Riemannian foliations and discuss open problems.
Intrinsic formulation of noncommutative geometry for quantum gravity.
We introduce a notion of the noncommutative integrability within a framework of contact geometry.
Researchers compute Connes-Chamseddine cycle on 6D manifolds using noncommutative integral.
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 class of noncommutative spaces studied is very rich. Non-anticommutative superspa…
Generalizes soft noncommutative schemes to flag varieties.
The paper introduces twins in Kähler and Sasaki geometry, generalizing known concepts.
In this letter we investigate some aspects of the noncommutative differential geometry based on derivations of the algebra of endomorphisms of an oriented complex hermitian vector bundle. We relate it, in a natural way, to the geometry of the underlying principal bundle and compute the cohomology of its complex of nonc…
A Riemannian geometry of noncommutative n-dimensional surfaces is developed as a first step towards the construction of a consistent noncommutative gravitational theory. Historically, as well, Riemannian geometry was recognized to be the underlying structure of Einstein's theory of general relativity and led to further…
Review of instantons in noncommutative gauge theories across 4, 6, and 8 dimensions.
We define some new invariants for 3-manifolds using the space of taut codim-1 foliations along with various techniques from noncommutative geometry. These invariants originate from our attempt to generalise Topological Quantum Field Theories in the Noncommutative geometry / topology realm.
It is emphasized that equivalent definitions of connections on modules over commutative rings are not so in noncommutative geometry.
The paper extends Lie bracket to noncommutative geometry using differential operators.
The work extends the A. Connes' noncommutative geometry to spaces with generic local anisotropy. We apply the E. Cartan's anholonomic frame approach to geometrical models and physical theories and develop the nonlinear connection formalism for projective module spaces. Examples of noncommutative generation of anholonom…
A simple equation explains standard model coupled to gravity.
We survey the geometry of Lagrange and Finsler spaces and discuss the issues related to the definition of curvature of nonholonomic manifolds enabled with nonlinear connection structure. It is proved that any commutative Riemannian geometry (in general, any Riemann--Cartan space) defined by a generic off--diagonal metr…
Study noncommutative deformations of Calabi-Yau threefolds.
We construct spectral triples in a sense of noncommutative differential geometry, associated with a Riemannian foliation on a compact manifold, and describe its dimension spectrum.
This is a sequel of \cite{Wang}, which provides a general formalism for this paper. We mainly investigate thoroughly a subclass of toric generalized Khler manifolds of symplectic type introduced by Boulanger in \cite{Bou}. We find torus actions on such manifolds are all \emph{strong Hamiltonian} in the sense …
Continuity of complex Monge-Ampère potentials on Kähler manifolds.
We study the noncommutative differential geometry of the algebra of endomorphisms of any SU(n)-vector bundle. We show that ordinary connections on such SU(n)-vector bundle can be interpreted in a natural way as a noncommutative 1-form on this algebra for the differential calculus based on derivations. We interpret the …
Develops noncommutative Cowen-Douglas theory for noncommuting operators.
In this paper we explain how to define "lower dimensional'' volumes of any compact Riemannian manifold as the integrals of local Riemannian invariants. For instance we give sense to the area and the length of such a manifold in any dimension. Our reasoning is motivated by an idea of Connes and involves in an essential …
Motivated by the local formulae for asymptotic expansion of heat kernels in spectral geometry, we propose a definition of Ricci curvature in noncommutative settings. The Ricci operator of an oriented closed Riemannian manifold can be realized as a spectral functional, namely the functional defined by the zeta function …
Defines and proves generalized noncommutative residue theorems for specific dimensions.
This paper has four main parts. In the first part, we construct a noncommutative residue for the hypoelliptic calculus on Heisenberg manifolds, that is, for the class of Heisenberg PsiDOs introduced by Beals-Greiner and Taylor. This noncommutative residue appears as the residual trace on integer order Heisenberg PsiDOs…
Anti-diagonal toric generalized Khler structures of symplectic type on a compact toric symplectic manifold were investigated in \cite{Wang2} . In this article, we consider \emph{general} toric generalized Khler structures of symplectic type, without requiring them to be anti-diagonal. Such a structu…
The main result of the paper is Egorov's theorem for transversally elliptic operators on compact foliated manifolds. This theorem is applied to describe the noncommutative geodesic flow in noncommutative geometry of Riemannian foliations.
This is a review of explicit computations of Connes distance in noncommutative geometry, covering finite dimensional spectral triples, almost-commutative geometries, and spectral triples on the algebra of compact operators. Several applications to physics are covered, like the metric interpretation of the Higgs field, …
We synthesize and extend the previous ideas about appearance of both noncommutative and Finsler geometry in string theory with nonvanishing B--field and/or anholonomic (super) frame structures \cite{vstring,vstr2,vnonc,vncf}. There are investigated the limits to the Einstein gravity and string generalizations containin…
A general definition of a bimodule connection in noncommutative geometry has been recently proposed. For a given algebra this definition is compared with the ordinary definition of a connection on a left module over the associated enveloping algebra. The corresponding curvatures are also compared.