Develops noncommutative Cowen-Douglas theory for noncommuting operators.
problem Exploring noncommutative analogues of classical Cowen-Douglas theory.
method Defining noncommutative Cowen-Douglas class using matricial joint eigenvalues and showing equivalence classes are determined by associated noncommutative vector bundles.
result Unitary equivalence class of a tuple in the noncommutative Cowen-Douglas class is determined by the equivalence class of its associated noncommutative vector bundle.
Study noncommutative Sobolev inequalities using quantum state metrics.
problem Establishing Sobolev inequalities in noncommutative settings.
method Generalizing monotone metrics in quantum states.
result Developed new matrix-valued Beckner inequalities.
Sheaves on graphs link to noncommutative geometry.
problem Exploring noncommutative geometry concepts on graphs.
method Sheaf theory and simplicial sets.
result Enhanced understanding of discrete noncommutative geometry.
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…
D-branes on noncommutative spaces mimic string theory, offering new insights into mirror symmetry.
problem Exploring noncommutative mirror symmetry through D-branes on noncommutative Calabi-Yau spaces.
method Constructing noncommutative ringed spaces from local resolutions, realizing D-branes as morphisms, and defining kinetic energy.
result Dynamical D-branes on noncommutative spaces can be described by a Polyakov-like action, suggesting a bridge between string theory and noncommutative geometry.
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 …
The paper extends Lie bracket to noncommutative geometry using differential operators.
problem Generalizing Lie bracket to noncommutative geometry.
method Antisymmetrizing compositions of vector fields and treating symbols of differential operators.
result Provided necessary and sufficient conditions for jet modules to represent differential operators.
In this paper we classify invariant noncommutative connections in the framework of the algebra of endomorphisms of a complex vector bundle. It has been proven previously that this noncommutative algebra generalizes in a natural way the ordinary geometry of connections. We use explicitely some geometric constructions us…
A (smooth) dynamical system with transformation group Tn is a triple (A,Tn,α), consisting of a unital locally convex algebra A, the n-torus Tn and a group homomorphism $α:\mathbb{T}^n\rightarrow\Aut(A)$, which induces a (smooth) continuous action of Tn on A. In this…
New framework for noncommutative Carrollian geometry using Lie-Rinehart pairs.
problem Developing a geometric framework for ultra-relativistic physics in noncommutative settings.
method Using ρ-Lie-Rinehart pairs to generalize Carrollian Lie algebroids to almost commutative geometry.
result Foundational principles of Carrollian geometry hold in almost commutative geometry.
Extends noncommutative deformations of holomorphic line bundles on complex tori and their mirror partners.
problem Noncommutative deformations of holomorphic line bundles on complex tori.
method Real nonformal deformation quantization and SYZ construction.
result Extended construction of noncommutative deformations of holomorphic line bundles.
We show the non-positivity of the Einstein-Hilbert action for conformal flat Riemannian metrics. The action vanishes only when the metric is constant flat. This recovers an earlier result of Fathizadeh-Khalkhali in the setting of spectral triples on noncommutative four-torus. Furthermore, computations of the gradient f…
In an earlier paper, we established a natural connection between the Baum-Connes conjecture and noncommutative Bloch theory, viz. the spectral theory of projectively periodic elliptic operators on covering spaces. We elaborate on this connection here and provide significant evidence for a fundamental conjecture in nonc…
We develop an invariant of knots that depends on a complex parameter t, describing a left ideal in the noncommutative torus. When the parameter is set equal to -1 we recover the A-polynomial of the knot. We relate the invariant to the colored Jones polynomials of the knot.
Constructs noncommutative spaces for D-branes on complex algebraic spaces.
problem Mathematical model for D-branes on noncommutative spaces.
method Toric geometry, Azumaya schemes, invertible sheaves.
result Embeds algebraic Calabi-Yau spaces into soft noncommutative schemes.
A well known result on pseudodifferential operators states that the noncommutative residue (Wodzicki residue) of a pseudodifferential projection vanishes. This statement is non-local and implies the regularity of the eta invariant at zero of Dirac type operators. We prove that in a filtered algebra the value of a proje…
Promotes spectral functionals to noncommutative fields and proves a theorem.
problem Spectral functionals on noncommutative fields and manifolds with boundary.
method Carries out promotion to spectral functionals, associates with noncommutative residue, and proves theorem.
result Proves Dabrowski-Sitarz-Zalecki type theorem for statistical de Rham Hodge operators on manifolds with boundary.
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…
Noncommutative geometry connects higher order connections to quantization.
problem Quantization in noncommutative geometry.
method Introducing natural linear differential operators and Spencer operators.
result Higher order connections are equivalent to quantization.
Researchers compute Connes-Chamseddine cycle on 6D manifolds using noncommutative integral.
problem Computing the Connes-Chamseddine cycle for 6D manifolds.
method Using noncommutative integral on 6D manifolds, they compute the cycle.
result The Connes-Chamseddine cycle on 6D manifolds is computed.
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…
We review previous work of Alain Connes, and its extension by the author, on some conformal invariants obtained from the noncommutative residue on even dimensional compact manifolds without boundary. Inspired by recent work of Yong Wang, we also address possible generalizations of these conformal invariants to the sett…
Defines and proves generalized noncommutative residue theorems for specific dimensions.
problem Defining and proving residue theorems for noncommutative geometry.
method Defined generalized noncommutative residue of Dirac operator; proved Kastler-Kalau-Walze type theorems.
result Validated Kastler-Kalau-Walze type theorems for 4D and 6D compact manifolds.
The paper explores noncommutative geometry of frame bundles using C*-algebras.
problem Understanding the noncommutative geometry of frame bundles.
method Using C*-algebras and unitary tensor functors, the paper constructs a free C*-dynamical system.
result Each C*-algebraic noncommutative principal SO(n)-bundle is uniquely determined by its associated noncommutative vector bundle.
This paper provides a description of an algebraic setting for the Lagrangian formalism over graded algebras and is intended as the necessary first step towards the noncommutative C-spectral sequence (variational bicomplex). A noncommutative version of integration procedure, the notion of adjoint operator, Green's formu…
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…
Develops Riemannian geometry for noncommutative super surfaces.
problem No specific problem stated; focuses on mathematical development.
method Introduces metric and connections on noncommutative super surfaces, showing compatibility and zero torsion under certain conditions.
result Noncommutative super surfaces have a well-defined Riemannian geometry with properties analogous to classical Riemannian geometry.
This paper is the the third part of a series of paper whose aim is to use of the framework of \emph{twisted spectral triples} to study conformal geometry from a noncommutive geometric viewpoint. In this paper we reformulate the inequality of Vafa-Witten \cite{VW:CMP84} in the setting of twisted spectral triples. This i…
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…
This is the second in a series of papers intended to set up a framework to study categories of modules in the context of non-commutative geometries. In \cite{mem} we introduced the basic DG category $\Pc_{\A^\bullet}$, the perfect category of $\A^\bullet$, which corresponded to the category of coherent sheaves on a com…
We introduce a framework for coverings of noncommutative spaces. Moreover, we study noncommutative coverings of irrational quantum tori and characterize all such coverings that are connected in a reasonable sense.
Intrinsic formulation of noncommutative geometry for quantum gravity.
problem Formalizing noncommutative differential geometry for quantum gravity.
method Geometric definitions and proofs of noncommutative Ricci curvatures and Bianchi identities.
result Quantum fluctuations and curvatures of (pseudo-) Riemannian metrics are renormalizable.
The paper constructs a noncommutative bracket on surface groups and proves it's Hamiltonian.
problem Noncommutative Hamiltonian structures on surface groups.
method Double quasi Poisson bracket construction and noncommutative r-matrix formalism. result Noncommutative Hamiltonian structures on cyclic spaces of unbased loops.
We study Finsler black holes induced from Einstein gravity as possible effects of quantum spacetime noncommutativity. Such Finsler models are defined by nonholonomic frames not on tangent bundles but on (pseudo) Riemannian manifolds being compatible with standard theories of physics. We focus on noncommutative deformat…
Derives Atiyah sequence for noncommutative bundles.
problem Deciding when ∗-automorphisms lift to compatible ones. method Derivation-based Atiyah sequence derivation.
result Validates existence of compatible lifts.
First, we review the notion of a Poisson structure on a noncommutative algebra due to Block-Getzler and Xu and introduce a notion of a Hamiltonian vector field on a noncommutative Poisson algebra. Then we describe a Poisson structure on a noncommutative algebra associated with a transversely symplectic foliation and co…
Generalizes soft noncommutative schemes to flag varieties.
problem Applying soft noncommutative schemes to flag varieties.
method Generalization via toric geometry and distinguished affine charts.
result Soft noncommutative schemes can be applied to flag varieties.
Review of instantons in noncommutative gauge theories across 4, 6, and 8 dimensions.
problem Understanding instantons in noncommutative gauge theories.
method Analysis of instantons in various dimensions, focusing on string theory and toric varieties.
result Geometric interpretations and applications of instantons in non-compact toric varieties.
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…
In this paper we prove a version of Connes' trace theorem for noncommutative tori of any dimension~n≥2. This allows us to recover and improve earlier versions of this result in dimension n=2 and n=4 by Fathizadeh-Khalkhali. We also recover the Connes integration formula for flat noncommutative tori of McDonal…
In this paper we produce several new invariants for CR and contact manifolds by looking at the noncommutative residue traces of various geometric projections. In the CR setting these operators arise from the Kohn-Rossi complex and include the Szegö projections on forms. In the contact setting they stem from the general…
The paper derives spectral (0,4)-tensor functionals using the noncommutative residue.
problem Deriving spectral (0,4)-tensor functionals on compact spin manifolds.
method Using four one-forms and the Dirac operator, the noncommutative residue is applied to even-dimensional compact spin manifolds.
result Spectral (0,4)-tensor functionals are extended to a general spectral triple.
We show that Calabi-Yau manifolds are emergent from the commutative limit of six-dimensional noncommutative Hermitian U(1) instantons. Therefore we argue that the noncommutative Hermitian U(1) instantons correspond to quantized Calabi-Yau manifolds.
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…
We present new classes of exact solutions with noncommutative symmetries constructed in vacuum Einstein gravity (in general, with nonzero cosmological constant), five dimensional (5D) gravity and (anti) de Sitter gauge gravity. Such solutions are generated by anholonomic frame transforms and parametrized by generic off…
Combines noncommutative geometry and spectral theory for new Weyl laws.
problem Developing new Weyl laws for noncommutative manifolds.
method Functional analysis, spectral theory, and Tauberian conditions.
result Generalizes and simplifies recent results on Weyl laws and integration formulas.
New bridge between diffeology and noncommutative geometry.
problem Connecting diffeology and noncommutative geometry.
method Embedding quasifolds into diffeology and associating C*-algebras.
result Morita classes of C*-algebras associated with diffeomorphic quasifolds.
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.