The paper explores Hermitian Clifford analysis and its connection to representation theory.
problem Determining if Hermitian Clifford operators are natural generalizations of orthogonal Dirac operators.
method Using representation theory of Lie groups to construct Dirac-type operators, but encountering issues with irreducibility of representations for Hermitian structures.
result The generalized gradient construction based on representation theory is the natural way to construct Dirac-type operators, but not for Hermitian Dirac operators due to representation issues.
Monogenic functions are basic to Clifford analysis. On Euclidean space they are defined as smooth functions with values in the corresponding Clifford algebra satisfying a certain system of first order differential equations, usually referred to as the Dirac equation. There are two equally natural extensions of these eq…
New proof of eta invariant results using hypoelliptic Laplacian and Clifford algebras.
problem Proving results on orbital integrals of eta invariants on compact locally symmetric spaces.
method Combining hypoelliptic Laplacian approach with Clifford algebras and probabilistic methods.
result Construction of proper Itô calculus for hypoelliptic diffusions.
Study uses geometric algebra to analyze credit cycles, revealing dangerous feedback loops.
problem Understanding and predicting dangerous feedback loops in credit cycles.
method Represent economic states as multi-vectors in Clifford algebra, focusing on bivector elements for rotational coupling.
result Geometric relationship between unemployment and credit contraction shifts from simple correlation to dangerous rotational dynamics during crises.
New Clifford-Weyl structures defined on conformal manifolds.
problem Understanding the geometry of even Clifford structures on conformal manifolds.
method Introduced Clifford-Weyl structures and showed conditions for their closure.
result Weyl structures are closed except in low-dimensional cases.
Based on \cite{DH94}, we introduce a bijective correspondence between first order differential calculi and the graph structure of the symmetric lattice that allows one to encode completely the interconnection structure of the graph in the exterior derivative. As a result, we obtain the Grassmannian character of the lat…
The paper examines smoothness in graded skew Clifford algebras.
problem Smoothness of graded skew Clifford algebras.
method Investigation of differential smoothness.
result Results on the differential smoothness of graded skew Clifford algebras.
The paper explores diffeological Clifford algebras and pseudo-bundles.
problem Constructing pseudo-bundles of diffeological Clifford algebras and modules.
method Using diffeological gluing to construct pseudo-bundles.
result Construction of pseudo-bundles of diffeological Clifford algebras and modules.
Study relates Finsler structures to Clifford bundles for flat metrics.
problem Relating Finsler structures to Clifford bundles for flat metrics.
method Examines extensions of Clifford bundles and Finsler type structures for flat metrics.
result Triangle map exists between Finsler structures constructed from metrics and 1-forms.
Complete classification of foliations on spheres from Clifford systems.
problem Classifying foliations on spheres from Clifford systems.
method Classification of homogeneous singular Riemannian foliations of spheres.
result Classification completed for foliations initiated by the second author.
Extends Kostant's results to symmetric pairs in Clifford algebras.
problem Analyzing k-invariants in Clifford algebras of symmetric pairs. method Proves Cartan theorem, transgression theorem, Harish-Chandra isomorphism, and Clifford algebra conjecture for relative case.
result Establishes a relative transgression theorem and Harish-Chandra isomorphism for Clifford algebras.
A Clifford-Wolf translation of a connected Finsler space is an isometry which moves each point the same distance. A Finsler space (M,F) is called Clifford-Wolf homogeneous if for any two points x1,x2∈M there is a Clifford-Wolf translation ρ such that ρ(x1)=x2. In this paper, we give a complete classifi…
The paper studies 4-qubit Clifford states and their properties.
problem Understanding the set and properties of 4-qubit Clifford states.
method Analyzing the 293760 4-qubit Clifford states, splitting them into 18 groups, and studying the action of CNOT gates and local gates.
result There are 293760 4-qubit Clifford states with specific entanglement entropies, and any pair can be connected with local gates and at most 3 CNOT gates.
Construct Clifford systems on Euclidean spaces and manifolds.
problem No specific problem stated; general Clifford systems construction.
method Inductive construction and adaptation to manifolds.
result Developments in octonionic geometry.
New minimal submanifolds in spheres share properties of the Clifford torus.
problem Finding new minimal surfaces in spheres.
method Analyzing properties of the Clifford torus and extending to other minimal submanifolds.
result More minimal submanifolds in spheres have helicoidal properties.
New symmetric Willmore tori emerge from Clifford torus in Berger spheres.
problem Finding new symmetric Willmore surfaces from Clifford torus.
method Applying bifurcation theory to estimate Morse index of Willmore surfaces.
result New symmetric Willmore tori emerge from Clifford torus.
A Clifford-Wolf translation of a connected Finsler space is an isometry which moves each point the sam distance. A Finsler space (M,F) is called Clifford-Wolf homogeneous if for any two point x1,x2∈M there is a Clifford-Wolf translation ρ such that ρ(x1)=x2. In this paper, we study Clifford-Wolf transl…
New theorem on solvable compact Clifford-Klein forms for certain homogeneous spaces.
problem Non-existence of solvable compact Clifford-Klein forms in reductive homogeneous spaces.
method Generalization of Benoist's theorem to 'very regular' embeddings of H into G.
result Proves non-existence for a specific class of homogeneous spaces.
Improved MoM estimator enhances classical shadows protocol for quantum measurements.
problem Efficient estimation of expectation values with reduced measurement shots.
method Modified median-of-means estimator with optimal constants and U-statistics.
result Improved performance of modified estimator for Clifford measurements.
Study on curvature tensors, discovering new Osserman tensors.
problem Investigate properties of curvature tensors and their relations.
method Introduce quasi-Clifford curvature tensors and analyze their properties.
result Discovered an Osserman curvature tensor not satisfying the duality principle.
Classifies compact Clifford-Klein forms for specific Lie algebras.
problem Classifying compact Clifford-Klein forms for given Lie algebra structures.
method Using Onishchik's results on semisimple Lie algebras, the paper classifies forms for triples (g,h,l).
result New examples of reductive homogeneous spaces with non-standard compact Clifford-Klein forms.
Researchers describe even Clifford structures on specific Grassmannians.
problem Understanding even Clifford structures on Grassmannians.
method Explicit description of structures on real, complex, and quaternionic Grassmannians.
result Explicit description of non-flat parallel even Clifford structures of ranks 8, 6, and 5.
The Clifford torus is uniquely identified as a Lagrangian self-shrinker in complex space.
problem Characterizing the Clifford torus as a Lagrangian self-shrinker in complex space.
method Analyzing the Clifford torus in C2 with specific curvature conditions. result The Clifford torus is the unique compact orientable Lagrangian self-shrinker in C2 with ∣A∣2≤2. In this paper, we study Clifford-Wolf translations of Finsler spaces. We first give a characterization of Clifford-Wolf translations of Finsler spaces in terms of Killing vector fields. In particular, we show that there is a natural correspondence between Clifford-Wolf translations and the Killing vector fields of cons…
The paper defines a new twistor space for Riemannian manifolds with even Clifford structures.
problem No specific problem stated; the focus is on a new mathematical structure.
method Introduced a new twistor space for Riemannian manifolds with even Clifford structures.
result Constructed almost complex structures on the twistor space for parallel even Clifford structures and proved integrability in some cases.
Compactify reductive homogeneous spaces and their Clifford-Klein forms.
problem Smooth compactification of reductive homogeneous spaces and their Clifford-Klein forms.
method Using Anosov representations and proper discontinuous actions of word hyperbolic groups.
result Topologically tame compactifications of Clifford-Klein forms.
Study even-Clifford structures on manifolds with large automorphism groups.
problem Classifying manifolds with large automorphism groups and even-Clifford structures.
method Classification and gap theorem for automorphism groups.
result Classification of simply connected manifolds with maximal automorphism groups.
The Clifford torus is unstable but rigid in mean curvature flow.
problem Stability and rigidity of the Clifford torus in mean curvature flow.
method Analysis of higher order phenomena, including entropy minimisation and infinitesimal deformations.
result The Clifford torus is locally unique as a self-shrinker for mean curvature flow.
The abstract constructs equivalences between complex Clifford modules and Lipschitz structures.
problem Classifying pseudo-Riemannian manifolds with specific structures.
method Mutually quasi-inverse equivalences between bundles of complex Clifford modules and reduced complex Lipschitz structures.
result A manifold admits a bundle of irreducible complex Clifford modules if it has a specific structure.
Constructs a model for differential KO-theory using Clifford modules.
problem Refining Atiyah and Singer's families index with differential structure.
method Builds a model using families of Clifford modules with superconnection.
result Affords a differential refinement of Atiyah and Singer's families index.
Improved bound on groups preserving Clifford parallelism in 4 dimensions.
problem Characterizing Clifford parallelism by automorphisms.
method Improving the bound on the dimension of groups preserving Clifford parallelism.
result Improved bound to 4 dimensions for groups preserving Clifford parallelism.
The paper explores connections between quaternionic and Cayley calibrations in dimensions 8 and 16.
problem Exploring connections between quaternionic and Cayley calibrations in dimensions 8 and 16.
method Starting from collections of 'Kähler 2-forms', the paper constructs canonical 4-forms and calibrated 4-planes in dimensions 8 and 16.
result Explicit formulas for canonical 4-forms ΦSpin(8) and ΦSpin(7)U(1) are derived, and their calibrated 4-planes are characterized. Novel analysis of neural networks using geometric algebra and convex optimization.
problem Understanding the inner workings of deep neural networks.
method Geometric (Clifford) algebra and convex optimization.
result Optimal weights are given by the wedge product of training samples.
A homogeneous space G/H is said to have a compact Clifford-Klein form if there exists a discrete subgroup D of G that acts properly discontinuously on G/H, such that the quotient space D\G/H is compact. When n is even, we find every closed, connected subgroup H of G = SO(2,n), such that G/H has a compact Clifford-Klein…
The geometry of nonholonomic bundle gerbes, provided with nonlinear connection structure, and nonholonomic gerbe modules is elaborated as the theory of Clifford modules on nonholonomic manifolds which positively fail to be spin. We explore an approach to such nonholonomic Dirac operators and derive the related Atiyah-S…
Unified study of surfaces using Clifford algebras.
problem Classifying immersed surfaces in various manifolds.
method Using Clifford algebras to construct formalism for immersed bilegendrian surfaces.
result Full classifications of immersed bilegendrian surfaces in the unit tangent bundle of the 3-sphere.
We provide a necessary condition for the existence of a compact Clifford-Klein form of a given homogeneous space of reductive type. The key to the proof is to combine a result of Kobayashi-Ono with an elementary fact that certain two different Clifford-Klein forms have the same cohomology ring. We give some examples, S…
Study on the structure groups of specific manifolds and their Spin properties.
problem Understanding the structure groups of almost even-Clifford Hermitian manifolds.
method Computing structure groups and determining Spin structures.
result Determined conditions for structure groups to lead to Spin structures.
Study pseudo-bundles of exterior algebras and their Clifford modules, addressing compatibility issues.
problem Compatibility issues in diffeological pseudo-bundles and their duals.
method Analysis of pseudo-bundles of exterior algebras, Clifford actions, and gluing conditions.
result A natural map ensuring commutativity of duals under gluing, also an isometry.
New proof of divisibility property for certain algebraic varieties.
problem Divisibility property for LQEL varieties.
method Construction of Clifford algebra representations to Severi varieties.
result New proof of Russo's Divisibility Property for LQEL varieties.
We compute the Bott-Morse Floer cohomology of the Clifford torus in $\CP^n$ with all possible spin-structures. Each spin structure is known to determine an orientation of the moduli space of holomorphic discs, and we analyze the change of orientation according to the change of spin structure of the Clifford torus. Also…
The paper extends matrix inequalities to various types of matrices.
problem Generalizing inequalities for different types of matrices.
method Extending known inequalities for real, complex, and quaternionic matrices.
result New inequalities for matrices in subspaces spanned by Clifford systems or algebras.
Our goal is to generalize the Choe-Hoppe helicoid and Clifford cones in Euclidean space. By sweeping out L indpendent Clifford cones in R2N+2 via the multi-screw motion, we construct minimal submanifolds in RL(2N+2)+1. Also, we sweep out the L-rays Clifford cone (introduced in Sectio…
Maps are shown to be Riemannian products with Ricci-flat fibers.
problem Understanding maps between manifolds and their geometric properties.
method Spin geometry and representation theory of curvature operators.
result Scalar-rigid maps are essentially Riemannian products of base and Ricci-flat fibers.
Non-trivial Clifford bundle from loop space tangent bundle.
problem Triviality obstruction of Clifford bundle on loop space.
method Constructing Clifford algebra bundle from loop space tangent bundle, showing non-triviality through Stiefel-Whitney and Pontrjagin classes.
result Clifford bundle is non-trivial, obstructed by manifold's Stiefel-Whitney and Pontrjagin classes.
We classify the algebraic curvature tensors which are both Osserman and complex Osserman in all but a finite number of exceptional dimensions.Information concerning the possible eigenvalue structures, which is provided by methods of algebraic topology, plays a central role in the analysis.
The Clifford group for 2 qubits is divided into 20 orbits, each with 4608 matrices.
problem Understanding the structure of the Clifford group for 2 qubits.
method Equivalence relation based on local Clifford gates and analysis of orbits.
result The Clifford group for 2 qubits is divided into 20 orbits, each with 4608 matrices.
No standard compact Clifford-Klein forms found for exceptional Lie groups.
problem Proving the non-existence of standard compact Clifford-Klein forms for exceptional Lie groups.
method Computer-aided approach, algorithmic methods for classifying semisimple subalgebras, and invariant calculations.
result Proves the non-existence of standard compact Clifford-Klein forms for homogeneous spaces of exceptional Lie groups.