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.
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.
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 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 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.
Survey on cohomology of exceptional spaces using Clifford structures.
problem Understanding cohomology of exceptional symmetric spaces.
method Construction of canonical octonionic Kähler forms and even Clifford structures.
result Describes primitive Betti numbers for most exceptional symmetric spaces.
New parallelisms of 3D projective space found and classified.
problem Classifying parallelisms of 3D projective space with specific automorphism groups.
method Generalized existing constructions to include oriented parallelisms and showed that only Clifford parallelism remains for higher-dimensional automorphism groups.
result Only Clifford parallelism remains for topological parallelisms with automorphism groups of dimension 3 or larger.
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 …
Study H-type foliations in sub-Riemannian geometry.
problem Understand and classify H-type foliations in sub-Riemannian geometry.
method Introduce and study H-type foliations, prove curvature bounds, and classify foliations with parallel structures.
result Prove curvature bounds and classify H-type foliations with parallel structures.
Unified characterization of special Riemannian holonomy groups using twisted pure spinors and Clifford monopoles.
problem Characterizing special Riemannian holonomy groups in a unified way.
method Introducing twisted pure spinors and Clifford monopole equations.
result Non-trivial solutions to Clifford monopole equations on manifolds with special Riemannian holonomy.
We provide explicit spinor representations for Clifford algebras.
problem Building explicit representations of Clifford algebras.
method Explicit construction of spinor modules and parallel spinor fields.
result Explicit spinor representations for all mixed signature Clifford algebras.
Geometric calculus introduced on pseudo-Riemannian manifolds without embedding.
problem Developing calculus on pseudo-Riemannian manifolds without embedding.
method Direct axiomatic approach to geometric calculus, paralleling general relativity.
result Full theory of differential calculus for vector, multivector, and tensor fields developed.
A generalized Clifford manifold is proposed in which there are coordinates not only for the basis vector generators, but for each element of the Clifford group, including the identity scalar. These new quantities are physically interpreted to represent internal structure of matter (e.g. classical or quantum spin). The …
We construct embedded closed minimal surfaces in the round three-sphere, resembling two parallel copies of the Clifford torus, joined by m^2 small catenoidal bridges symmetrically arranged along a square lattice of points on the torus.
The paper explores parallelisms and spreads in 3D projective space.
problem Characterizing parallelisms and spreads in PG(3,R).
method Introducing topological parallelisms, analyzing their properties and relationships with spreads.
result There are more oriented parallelisms than ordinary parallelisms, and the automorphism group is SO(3).
Explains conformal maps and holomorphic structures for surfaces in higher-dimensional Euclidean spaces.
problem Explains conformal maps and holomorphic structures for surfaces in higher-dimensional Euclidean spaces.
method Uses Clifford algebra and spin transforms to explain conformal maps and holomorphic structures.
result Calculates the degree of the spinor bundle associated with a conformal immersion and defines analogues of polar and bipolar surfaces.
New minimal surface doublings of Clifford Torus improve bounds on minimal surfaces in S^3.
problem Proving bounds on minimal surfaces in S^3 with fixed genus.
method Applying a general theorem to produce new minimal doublings of the Clifford Torus, using min-max methods, and verifying Yau's conjecture.
result Improved quadratic lower bound for the number of embedded minimal surfaces in S^3 with prescribed genus.
Reviews interactions between Spin(9) and octonionic geometries.
problem Understanding the role of Spin(9) in octonionic geometry.
method Analyzes canonical 8-forms, vector fields, Hopf fibrations, and manifolds.
result Discovers new insights into the geometry of octonionic Hopf fibrations.
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.
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.
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.
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.
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 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. The paper classifies obstructions to real Clifford module bundles and their relation to spin structures.
problem Topological obstructions to real Clifford module bundles over pseudo-Riemannian manifolds.
method Analyzes real Lipschitz structures and their relation to spin structures.
result Classifies real Lipschitz structures in all dimensions and signatures.
Estimates eigenvalues of modified Dirac operators with multi-forms.
problem Estimating eigenvalues of modified Dirac operators with multi-forms.
method Analyzes eigenvalues of multi-form modified Dirac operators constructed from a standard Dirac operator.
result Provides estimates for eigenvalues of modified Dirac operators with multi-forms.
This dissertation explores Clifford bundles and spinor fields in geometric and algebraic contexts.
problem Understanding spinor fields and their classification in geometric frameworks.
method Combines algebraic and geometric approaches to study Clifford structures on bundles and spinor fields.
result Identifies new spinor field classes in warped flux compactifications.
Spin geometry reviewed with applications in physics.
problem None explicitly stated, but related to spin geometry and its applications.
method Definitions of spin geometry, Clifford algebras, and spinors; discussion of differential operators, twistor and Killing spinors; holonomy classification; construction of symmetry operators and extended superalgebras.
result Construction of extended superalgebras and methods to find solutions of Seiberg-Witten equations.
Topological theory for qLDPC codes enables non-Clifford gates and magic state injection.
problem Fault-tolerant quantum computation in qLDPC codes with non-Clifford gates and magic state resources.
method Developed a topological theory using simplicial or CW complex structures and deformation retraction.
result Achieved non-Clifford gates and magic state injection in qLDPC codes with constant rate and polynomial distance.
Paper analyzes U(n)-structures and their minimal left ideals.
problem Understanding U(n)-structures and their minimal left ideals. method Identifying U(n)-structures with minimal left ideals via induced Kahler polynomial. result Established link between U(n)-structures and minimal left ideals for Clifford algebras. Class lecture notes at a beginning graduate level on the mathematical background needed to understand classical gauge theory. Covers group actions, fiber bundles, principal bundles, connections, gauge transformations, parallel transport, curvature, covariant derivatives, pseudo-riemannian manifolds, lagrangians, cliffo…
Paper connects geometric structures to algebra in high dimensions.
problem Understanding geometric structures in high dimensions.
method Relating minimal left ideals on Clifford algebras to geometric structures.
result Established a connection between algebraic and geometric properties.
An almost Clifford and an almost Cliffordian manifold is a G--structure based on the definition of Clifford algebras. An almost Clifford manifold based on $\mathcal O:= \cc l (s,t)$ is given by a reduction of the structure group GL(km,R) to GL(m,O), where k=2s+t and m∈N. An…
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.
Introduces a new geometric product for differential forms.
problem Constructing a Clifford algebra from differential forms.
method Introduces and revisits the Graf product with a new framework.
result Provides a structure for constructing a Clifford algebra.
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 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.
We propose a new framework for constructing geometric and physical models on nonholonomic manifolds provided both with Clifford -- Lie algebroid symmetry and nonlinear connection structure. Explicit parametrizations of generic off-diagonal metrics and linear and nonlinear connections define different types of Finsler, …
We provide a characterization of the Clifford Torus in S3 via moving frames and contact structure equations. More precisely, we prove that minimal surfaces in S3 with constant contact angle must be the Clifford Torus. Some applications of this result are then given, and some examples are discussed.
Sharp pinching theorem for submanifolds in spheres.
problem Characterizing submanifolds in spheres based on curvature bounds.
method Conformal method of Fischer-Colbrie, Shen & Ye and Catino, Mastrolia & Roncoroni.
result Complete submanifolds with specific curvature bounds are either totally geodesic or Clifford tori/Veronese surfaces.
In this paper we examine a new class of five dimensional (5D) exact solutions in extra dimension gravity possessing Lie algebroid symmetry. The constructions provide a motivation for the theory of Clifford nonholonomic algebroids elaborated in Ref. hep-th/0501217. Such Einstein-Dirac spacetimes are parametrized by gene…
We present an introduction to the geometry of higher order vector and co--vector bundles (including higher order generalizations of the Finsler geometry and Kaluza--Klein gravity) and review the basic results on Clifford and spinor structures on spaces with generic local anisotropy modeled by higher order nonlinear con…
We give an upper bound for the rank r of homogeneous (even) Clifford structures on compact manifolds of non-vanishing Euler characteristic. More precisely, we show that if r=2a⋅b with b odd, then r≤9 for a=0, r≤10 for a=1, r≤12 for a=2 and r≤16 for a≥3. Moreover, we describe t…
The paper describes a new geometric structure for general Clifford algebras.
problem Understanding the geometry of general Clifford algebras.
method Introducing a new geometric structure called a mixed structure for Cl(r,s).
result There are multiple creation/annihilation operator models for a given Clifford algebra.
Novel CG-EGNNs learn equivariant functions from Clifford algebras.
problem Lack of equivariance in high-order graph neural networks.
method Integrates high-order local structures with Clifford algebras for equivariant learning.
result CG-EGNNs outperform previous methods on various benchmarks.
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.
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.