Generalizes equivariance and convolution to compact groups for neural networks.
problem Ensuring equivariance in neural networks for various domain actions.
method Representation theory and noncommutative harmonic analysis.
result Convolution is necessary and sufficient for equivariance to compact group actions.
We review basic notions and methods of noncommutative geometry and their applications to analysis and geometry on foliated manifolds.
Formulae for Dixmier trace and noncommutative residue on compact manifolds.
problem Calculating traces and residues for pseudo-differential operators on compact manifolds.
method Using global symbols, Fourier analysis, representation theory, and pseudo-differential calculus.
result Formulae for Dixmier trace and noncommutative residue on compact manifolds.
Study differential and integral calculus on noncommutative C*-algebras.
problem Develop calculus on noncommutative spaces.
method Formal smooth structure on nonpure states of C*-algebras.
result Prove Stokes' theorem in both commutative and noncommutative settings.
Simpler neural network for spherical images using Clebsch-Gordan transforms.
problem Learning spherical images rotation invariantly.
method Clebsch-Gordan transform for nonlinearity, avoiding repeated Fourier transforms.
result Improved performance compared to previous methods.
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 …
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.
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 coverings of irrational quantum tori.
problem Characterize noncommutative coverings of irrational quantum tori.
method Developed a framework for noncommutative coverings and studied irrational quantum tori.
result Characterized all connected noncommutative coverings of irrational quantum tori.
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.
Curvature defined in noncommutative geometry for curved spaces.
problem Defining curvature in noncommutative geometry for curved spaces.
method Using spectral geometry and heat kernel asymptotic expansions.
result New ways to define local curvature invariants for noncommutative Riemannian spaces.
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.
Erdős introduced the noncommuting graph, in order to study the number of commuting elements in a finite group. Despite the use of combinatorial ideas, his methods involved several techniques of classical analysis. The interest for this graph is becoming relevant in the last years for various reasons. Here we deal with …
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…
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…
New invariants for 3-manifolds from foliations and noncommutative geometry.
problem Creating new invariants for 3-manifolds.
method Using taut codim-1 foliations and noncommutative geometry techniques.
result Definition of new invariants for 3-manifolds.
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.
The paper proves a Connes trace theorem for curved noncommutative tori.
problem Recovering scalar curvature in curved noncommutative tori.
method Proving a version of Connes' trace theorem for noncommutative tori of any dimension.
result Establishes a curved version of Connes' integration formula for scalar curvature.
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 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…
The paper constructs Laplace-Beltrami operators on noncommutative tori.
problem Developing Laplace-Beltrami operators for noncommutative tori.
method Construction of Laplace-Beltrami operators with consideration of non-trivial modular automorphisms.
result Laplace-Beltrami operators on noncommutative tori have properties similar to those on ordinary Riemannian manifolds.
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…
Unveils fermions' geometric nature in the Standard Model as noncommutative forms.
problem Understanding the geometric structure of fermions in the Standard Model.
method Uses noncommutative geometry to represent fermion multiplet as de Rham forms.
result Fermions in the Standard Model are represented as noncommutative de Rham forms.
Develops an L^p theory for Dolbeault-Dirac operators on compact Kähler manifolds.
problem Analyzing Dolbeault-Dirac operators on compact Kähler manifolds with Banach space coefficients.
method Establishes an L^p theory for Dolbeault-Dirac operators, proving bisectoriality, H^\infty functional calculus, and Gaffney-type estimates.
result Identifies the index of the associated Fredholm operator with the holomorphic Euler characteristic, independent of p.
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.
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…
Defines Ricci curvature in noncommutative geometry.
problem No specific problem stated; focuses on definition and computation.
method Proposes a definition using spectral functionals and zeta functions.
result Explicit computation of Ricci density for conformally flat noncommutative tori.
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.
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.
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…
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.
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 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.
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…
Flat connections on bundles from noncommutative representations.
problem Noncommutative analog of flat connections.
method Linear representations of noncommutative fundamental groups.
result Finite noncommutative coverings and flat connections.
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.
New spectral functionals related to noncommutative residue and torsion Dirac operators.
problem Spectral functionals and Dirac operators with torsion.
method Noncommutative residue and Dirac operators with torsion.
result Extension of spectral functionals to noncommutative realm with torsion.
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.
Together with collaborators, we introduced a noncommutative Riemannian geometry over Moyal algebras and systematically developed it for noncommutative spaces embedded in higher dimensions in the last few years. The theory was applied to construct a noncommutative version of general relativity, which is expected to capt…
The paper develops a new probabilistic framework for denoising diffusion models using free entropy and stochastic analysis.
problem Developing a mathematical framework for denoising diffusion models in noncommutative settings.
method Formulating diffusion and reverse processes governed by operator-valued stochastic dynamics, using tools from free stochastic analysis.
result Establishing an information-geometric link between entropy production, transport, and deconvolution.