Complex Cayley Grassmannian is a singular variety with smooth singular locus.
problem Characterize the singularities of the Cayley Grassmannian.
method Defined a torus action and used it to prove singularity and cohomology properties.
result Singular locus is smooth and has the same cohomology as CP5. Study harmonic maps into Cayley plane using twistor theory.
problem Harmonic maps from Riemann surfaces into the Cayley plane.
method Twistor theory applied to nilconformal harmonic maps, using Lie algebra classifications.
result Constructs harmonic maps into the Cayley plane using submanifold techniques.
We describe an action of U(1)×Sp(1) on HP7 that allows to obtain a Z2-quotient of CAYLEY (the Grassmannian of oriented 4-planes in R8 closed under the three-fold cross product) by quaternionic Kähler reduction.
Cayley cones in the octonions O that are ruled by oriented 2-planes are equivalent to pseudoholomorphic curves in the Grassmannian of oriented 2-planes G(2,8). The well known twistor fibration G(2,8)−>S6 is used to prove the existence of immersed higher-genus pseudoholomorphic curves in $\gro$. Equivale…
This paper proves area-minimizing cones over products of Grassmannian manifolds.
problem Proving area-minimizing cones over products of Grassmannian manifolds.
method Using Hermitian orthogonal projectors and carefully computing the Jacobian.
result Cone over minimal products of Grassmannian manifolds are area-minimizing.
The paper studies foliations on homogeneous spaces and identifies specific foliations.
problem Understanding foliations on homogeneous varieties.
method Equivariant techniques applied to rational homogeneous spaces.
result Identified specific foliations on various homogeneous spaces.
This paper proves area-minimizing cones over Grassmannian manifolds.
problem Determine if cones over Grassmannian manifolds are area-minimizing.
method Detailed descriptions of embedding maps using Hermitian orthogonal projectors, re-proving area-minimization using Lawlor's Curvature Criterion.
result All cones over Grassmannian manifolds are area-minimizing except for oriented real Grassmannians.
We study curvature-adapted submanifolds of general symmetric spaces. We generalize Cartan's theorem for isoparametric hypersurfaces of spheres and Wang's classification of isoparametric Hopf hypersurfaces in complex projective spaces to any compact symmetric space. Our second objective is to investigate such hypersurfa…
We complete the classification of supersymmetric configurations of two M5-branes, started by Ohta and Townsend. The novel configurations not considered before are those in which the two branes are moving relative to one another. These configurations are obtained by starting with two coincident branes and Lorentz-transf…
Study on stability of Einstein metrics on symmetric spaces.
problem Stability of Einstein-Hilbert functional on compact symmetric spaces.
method Classification of irreducible representations and use of Casimir eigenvalues.
result Proves stability of Einstein metrics on quaternionic and Cayley projective plane, instability on other quaternionic Grassmannians.
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.
Quadratic Killing tensors are not always decomposable on certain symmetric spaces.
problem Characterizing when quadratic Killing tensors are decomposable on symmetric spaces.
method Analyzing Killing tensor fields on spaces of constant curvature and higher rank.
result Not all quadratic Killing tensors are decomposable on quaternionic projective spaces and Cayley projective plane.
Study of Hitchin map on specific Higgs bundles.
problem Understanding the Hitchin map on even very stable Higgs bundles.
method Defined even very stable Higgs bundles and studied the Hitchin map on their upward flows.
result Classification of type (1,...,1) examples and their relation to root systems.
Deformations of singular Cayley submanifolds studied.
problem Constructing fibrations of compact Spin(7) manifolds.
method Deformation theory of conically singular and asymptotically conical Cayley submanifolds.
result Detailed description of the deformation theory.
Study on deformations of singular submanifolds in complex geometry.
problem Deformation theory of conically singular Cayley submanifolds.
method Proved expected dimension of moduli space and compared complex and Cayley deformations.
result Moduli space is smooth for 2D complex submanifolds of Calabi-Yau 4-folds.
Desingularizes conically singular Cayley submanifolds.
problem Constructing fibrations of compact Spin(7) manifolds by Cayley submanifolds.
method Desingularization through gluing rescaled asymptotically conical submanifolds.
result Conically singular Cayley submanifolds can be desingularized.
Study of Cayley structures linked to differential equations.
problem Understanding Cayley structures and their differential equations.
method Defined Cayley structures, introduced extensions, solved equivalence problem.
result Found fundamental invariants and examples of Cayley structures.
Classifies linear embeddings of grassmannians and ind-grassmannians.
problem Understanding linear embeddings of grassmannians and ind-grassmannians.
method Classification through isomorphism of Picard groups and direct limits.
result Most linear embeddings of grassmannians are equivariant.
The paper examines complex and para-complex structures on pseudo-Riemannian spheres.
problem Exploring non-integrable Cayley structures on pseudo-Riemannian spheres.
method Study of Cayley structures on six-dimensional pseudo-Riemannian spheres.
result Existence of complex and para-complex structures on pseudospheres.
Third in a series, this paper constructs non-trivial Cayley fibrations with conical singularities.
problem Constructing non-trivial Cayley fibrations with conical singularities.
method Using gluing methods and stability results for weak and usual fibrations.
result Construction of examples of Cayley fibrations on twisted connected sum G2 manifolds. The study examines discrete curvature notions on Cayley graphs of certain groups.
problem Understanding curvature in discrete settings for various groups.
method Introduced Right Angled Artin-Coxeter Hybrids (RAACHs) and derived curvatures of Cayley graphs.
result Addition of relators does not decrease weighted curvatures of Cayley graphs.
Efficiently optimizes CNN and RNN parameters on Stiefel manifold.
problem Computational expense in optimizing orthonormal matrices on Stiefel manifold.
method Cayley transform for efficient retraction and vector transport on Stiefel manifold.
result Cayley SGD and ADAM achieve faster convergence and less training time.
Study Cayley fibrations on Bryant-Salamon Spin(7) manifolds.
problem Investigate Cayley fibrations on specific Spin(7) manifolds. method Analyze invariant Cayley fibrations for each SU(2) action. result Explicitly describe the fibres of Cayley fibrations.
Given a Kaehler manifold of complex dimension 4, we consider submanifolds of (real) dimension 4, whose Kaehler angles coincide. We call these submanifolds Cayley. We investigate some of their basic properties, and prove that (a) if the ambient manifold is a Calabi-Yau, the minimal Cayley submanifolds are just the Cayle…
Eastwood and Ezhov generalized the Cayley surface to the Cayley hypersurface in each dimension, proved some characteristic properties of the Cayley hypersurface and conjectured that a homogeneous hypersurface in affine space satisfying these properties must be the Cayley hypersurface. We will prove this conjecture when…
The study introduces Cayley--Abels--Rosendal graphs for Polish groups.
problem Understanding the structure of Polish groups through graph theory.
method Developing Cayley--Abels--Rosendal graphs and applying them to Polish groups.
result Groups with Cayley--Abels--Rosendal graphs are topological analogues of finitely generated groups.
Quasi-transitive graphs quasi-isometric to planar graphs can be upgraded to Cayley graphs.
problem Quasi-transitive graphs quasi-isometric to planar graphs need to be upgraded to Cayley graphs.
method Upgrading a planar graph to a Cayley graph.
result Quasi-transitive graphs quasi-isometric to planar graphs can be upgraded to Cayley graphs.
The study finds Lie algebra formulae and classifies polar actions on a hyperbolic plane.
problem Finding Lie algebra formulae and classifying actions on hyperbolic planes.
method Using octonions and triality, explicit Lie brackets were found for Lie algebras of isometry groups.
result Explicit formulae for Lie brackets of f4 and f4∗ Lie algebras. The Monge-Cayley-Salmon theorem is extended for ruled submanifolds.
problem Generalization of a classical theorem for ruled submanifolds.
method Elementary geometric measure theory.
result Generalization of the Monge-Cayley-Salmon theorem proved.
The Cayley graph of quandles reveals structural properties and is studied for various classes.
problem Investigating structural properties of Cayley graphs of quandles.
method Analyzing Cayley graphs for different quandle classes and proving properties.
result Connected components of Cayley graphs of Alexander quandles correspond to cosets of specific subgroups.
Introduces Lorentzian Cayley form solving geometric puzzle.
problem Solving geometric puzzle in 4D Lorentzian geometry.
method Introduces complex Cayley forms and Lorentzian Cayley form.
result Lorentzian Cayley form solves geometric puzzle.
Study shows only two topological configurations for Spin(7)-manifold fibrations, ruling out smooth Cayley fibrations.
problem Understanding smooth fibrations of compact Spin(7)-manifolds by Cayley submanifolds.
method Geometric and topological constraints from Spin(7)-structure, spinnability criterion, gauge-theoretic input.
result Rules out smooth Cayley fibrations on all known compact torsion-free Spin(7)-manifolds.
Functions of several octonion variables are investigated and integral representation theorems for them are proved. With the help of them solutions of the ∂~-equations are studied. More generally functions of several Cayley-Dickson variables are considered. Integral formulas of the Martinelli-Bochner,…
The paper studies properties of Artin monoid Cayley graphs and their quasi-isometry to Deligne complexes.
problem Investigate properties of Artin monoid Cayley graphs.
method Show quasi-isometry to modified Deligne complex, address infinite diameter conjecture.
result Prove conjecture about infinite diameter for Artin groups containing specific subgroups.
We exhibit a family of homogeneous hypersurfaces in affine space, one in each dimension, generalising the Cayley surface.
Study proves existence of precotangent bundles for Grassmannians.
problem Existence of precotangent bundles for Grassmannians.
method Proof for Grassmannians of reflexive Banach spaces and p-restricted Grassmannians of polarized Hilbert space. result Existence of bundle predual to tangent bundle (precotangent bundle).
The paper finds inequalities in Grassmannian geometry.
problem Understanding geometric properties of Grassmannians.
method Analyzes inequalities for elements in Grassmannians.
result Law of Cosines and geodesic triangle inequalities.
Following an idea of Dadok, Harvey and Lawson, we apply the triality property of SO(8) to study the comass of certain self-dual 4-forms on R^8. In particular, we prove that the Cayley 4-form has comass 1 and that any self-dual 4-form realizing the maximal Wirtinger ratio is SO(8)-conjugate to the Cayley 4-form. We also…
We prove an existence theorem for Spin(7)-instantons, which are highly concentrated near a Cayley submanifold; thus giving a partial converse to Tian's foundational compactness theorem. As an application, we show how to construct Spin(7)-instantons on Spin(7)-manifolds with suitable local K3 Cayley fibrations. This rec…
The Cayley hyperbolic space minimizes volume entropy among finite-volume metrics.
problem Volume entropy rigidity in Cayley hyperbolic spaces.
method Repairing a gap in the proof of volume entropy rigidity theorem.
result Cayley hyperbolic space minimizes volume entropy.
This paper develops graph theory for racks and quasigroups.
problem Characterizing and realizing right quasigroups and related structures.
method Study of graph markings, Schreier graphs, and Cayley graphs.
result All right quasigroups are realizable by specific types of graphs.
Groups can act on 3-manifolds if their Cayley complex can embed in specific types of 3-manifolds.
problem Understanding group actions on 3-manifolds.
method Proving groups admit actions on 3-manifolds if their Cayley complexes can embed in specific 3-manifolds.
result Groups with embeddable Cayley complexes can act on four specific types of 3-manifolds.
Study complex deformations of compact complex surfaces in Calabi-Yau four-folds.
problem Explaining why complex and Cayley deformations of a compact complex surface are the same.
method Study complex deformations of compact complex submanifolds of Calabi-Yau manifolds.
result Prove that the moduli space of complex deformations of any compact complex embedded submanifold of a Calabi-Yau manifold is a smooth manifold.
Curious structure of special orthogonal, unitary, and symplectic groups as products of Grassmannians discovered.
problem Understanding the structure of special orthogonal, unitary, and symplectic groups.
method Expressing these groups as products of Grassmannians realized as involution matrices.
result Special orthogonal, special unitary, and symplectic groups can be expressed as products of their corresponding Grassmannians.
The paper explores the geometry of Lagrangian Grassmannians and their connection to PDEs.
problem Understanding the geometric properties of Lagrangian subspaces.
method Thorough review of geometric properties and their relation to PDEs.
result Hypersurfaces in the Lagrangian Grassmannian correspond to second-order PDEs.
Constructs explicit p-harmonic functions on Grassmannians and flag manifolds.
problem Finding proper p-harmonic functions on Grassmannians and flag manifolds. method Using the method of eigenfamilies to construct explicit functions.
result Explicit complex-valued proper p-harmonic functions on compact real Grassmannians and non-descending functions on real flag manifolds. Using the Plucker map between grassmannians, we study basic aspects of classic grassmannian geometries. For `hyperbolic' grassmannian geometries, we prove some facts (for instance, that the Plucker map is a minimal isometric embedding) that were previously known in the `elliptic' case.
Grassmannian sigma models extend Gross-Neveu model formulations.
problem Understanding sigma models on Grassmannian targets.
method Chiral Gross-Neveu model formulations for orthogonal and symplectic Grassmannians.
result One-loop β-functions proportional to dual Coxeter numbers.