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.
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 proves rigidity and vanishing of geometric indices on specific manifolds.
problem Indices of twisted Dirac operators on specific manifolds.
method Proves rigidity and vanishing of indices on almost even-Clifford Hermitian manifolds with circle actions.
result Proves rigidity and vanishing of indices for the specified manifolds.
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.
New operators generalize Michelsohn's on almost Hermitian manifolds.
problem Generalizing differential operators to almost Hermitian manifolds.
method Introducing two differential operators on sections of the complex Clifford bundle over compact almost Hermitian manifolds.
result Surprising Kähler-like symmetries in the kernel of the Laplacians of these operators.
Researchers solve the Calderón problem for fractional Dirac operators.
problem Determining the metric and structure from boundary measurements.
method Analyzing the fractional Dirac operator on vector bundles.
result The Calderón problem is solved uniquely for the fractional Dirac operator.
We introduce the notion of even Clifford structures on Riemannian manifolds, a framework generalizing almost Hermitian and quaternion-Hermitian geometries. We give the complete classification of manifolds carrying parallel even Clifford structures: Kähler, quaternion-Kähler and Riemannian products of quaternion-Kähler …
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 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.
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.
The paper examines sequences of solutions to Seiberg-Witten systems in 4D manifolds.
problem Analyzing sequences of solutions to Seiberg-Witten systems in 4D manifolds.
method Investigates the behavior of sequences of solutions to Seiberg-Witten-like equations for Hermitian connections and spinor sections.
result Provides insights into the behavior of sequences of solutions to Seiberg-Witten systems in 4D manifolds.
Study perturbs Dirac operators in any dimension, focusing on Majorana fermions.
problem Understanding perturbations of Dirac operators in various dimensions.
method Analyzes canonical perturbations of Dirac operators on Hermitian Clifford modules.
result Characterizes the low-energy spectrum of these operators on complete surfaces.
The fourth Severi variety has a special geometric structure related to parallel Clifford structures.
problem Characterizing the geometric structure of the fourth Severi variety.
method Explicit construction of a sub-bundle of End(TM) and associated differential forms.
result A canonical differential 8-form on the fourth Severi variety represents a generator of its cohomology ring.
Introduces Plücker coordinates for a complex projective octonion plane, solving an overdetermined system of relations.
problem Understanding the complex projective octonion plane and its quotient space EIII.
method Introduces Plücker coordinates and uses Clifford algebra to solve the overdetermined system of relations.
result Shows that EIII can be decomposed into F4-orbits and provides detailed analysis near the subvariety X∞.
New non-semisimple Ising anyons enable robust universal quantum computation.
problem Limitation of semisimple theories in universal topological quantum computation.
method Developed non-semisimple Ising anyon model with new anyon types indexed by α. result Robust universality of braiding persists over an open interval of α. We discuss algebraic properties for the symbols of geometric first order differential operators on almost Hermitian manifolds and Kähler manifolds. Through study on the universal enveloping algebra and higher Casimir elements, we know algebraic relations for the symbols like the Clifford algebra. From the relations, we…
New method classifies special Vinberg cones of rank 4.
problem Classifying special Vinberg cones of rank 4.
method Using Clifford Nil-algebras and directed acyclic graphs.
result Explicit classification of rank 4 special Vinberg cones.
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.
The present paper is a short survey on the mathematical basics of Classical Field Theory including the Serre-Swan' theorem, Clifford algebra bundles and spinor bundles over smooth Riemannian manifolds, Spin^C-structures, Dirac operators, exterior algebra bundles and Connes' differential algebras in the commutative case…
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.
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 proves conditions for the triviality of L2-harmonic forms on Riemannian manifolds.
problem Conditions for the triviality of L2-harmonic forms on Riemannian manifolds. method Study of a covariant Schrödinger operator HX,V and its L2-kernel. result Sufficient conditions for the triviality of the L2-kernel of HX,V. Flat Hermitian Lie algebras are always Kähler.
problem Classifying Lie groups with Hermitian structures that are flat.
method Analysis on the Hermitian geometry of 2-step solvable Lie groups.
result Flat Hermitian Lie algebras are Kähler.
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.
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.