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.
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.
Let M be an 8-manifold with a Spin(7)-structure. We first show that closed Cayley submanifolds of M form a smooth moduli space for a generic Spin(7)-structure. Then we study the deformations of a compact, connected Cayley submanifold X of M with non-empty boundary contained in a given submanifold W of M such that X and…
A nearly parallel G2-manifold Y is a Riemannian 7-manifold whose cone C(Y)=R>0×Y has the holonomy group contained in Spin(7). In other words, it is a spin 7-manifold with a real Killing spinor. We have a special class of calibrated submanifolds called Cayley submanifolds in C(Y).…
We study the deformations of an asymptotically cylindrical Cayley submanifold inside an asymptotically cylindrical Spin(7)-manifold. We prove an index formula for the operator of Dirac type that arises as the linearisation of the deformation map and show that if the Spin(7)-structure is generic, then there are no obstr…
Study third order Einstein deformations for Kähler-Einstein metrics on compact manifolds.
problem Existence of non-trivial Einstein deformations of Kähler metrics.
method Explicitly determined the obstruction to third order Einstein deformation and formulated it in terms of polynomial identities.
result Third order integrability for the Einstein equation is equivalent to Maurer-Cartan type equations and polynomial identities.
The paper studies deformations of calibrated subbundles in special holonomy manifolds.
problem Deforming calibrated subbundles in noncompact manifolds of special holonomy.
method Twisting calibrated subbundles by special sections and deriving conditions for deformations to remain calibrated.
result Twisting conormal bundles of Lagrangian submanifolds in T∗Sn by 1-forms does not provide new examples. Associative submanifolds A in nearly parallel G2-manifolds Y are minimal 3-submanifolds in spin 7-manifolds with a real Killing spinor. The Riemannian cone over Y has the holonomy group contained in Spin(7) and the Riemannian cone over A is a Cayley submanifold. Infinitesimal deformations of associat…
A long-standing open problem in systolic geometry asks whether a Riemannian metric on the real projective space whose volume equals that of the canonical metric, but is not isometric to it, must necessarily carry a periodic geodesic of length smaller than π. A contact-geometric reformulation of systolic geometry and th…
Defines a volume functional for Hermitian connections on manifolds, proving its properties and connections.
problem Defining and analyzing volume functionals for Hermitian connections on manifolds.
method Introduces the line bundle mean curvature flow and relates it to deformed connections and special submanifolds.
result Proves the mirror equality for mSpin(7)-dDT connections and deduces their properties. 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.
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.
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…
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.
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 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.
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…
By a theorem of Mclean, the deformation space of an associative submanifold Y of an integrable G_2 manifold (M,φ) can be identified with the kernel of a Dirac operator D:Ω^{0}(ν) -->Ω^{0}(ν) on the normal bundle νof Y. Here, we generalize this to the non-integrable case, and also show that the deformation space becomes…
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.
Formulates integrals for hypersphere arrangements using cohomology and Cayley-Menger determinants.
problem Formulating integrals for hypersphere arrangements.
method Using cohomology and Cayley-Menger determinants.
result Explicit representation and variational formula of integrals.
Alternative definition of dDT connections for Spin(7) manifolds.
problem Defining deformed Donaldson-Thomas connections for Spin(7) manifolds.
method Real Fourier-Mukai transform applied to compute new connections.
result Alternative definition of dDT connections is compatible with existing connections.
New unitary RNN architecture using complex Cayley transform outperforms existing methods.
problem Vanishing or exploding gradient problem in RNNs.
method Developed a unitary RNN architecture based on a complex scaled Cayley transform.
result scuRNN achieves comparable or better results than existing unitary RNNs.
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.
New proof shows surface group Cayley graphs have infinite diameter.
problem Infinite diameter of Cayley graphs for surface groups.
method New proof using mapping class group orbits.
result Cayley graphs have infinite diameter for certain generating sets.
Study of lamplighter group's Cayley graph spectrum and random Schrödinger operators.
problem Characterize the spectrum of Cayley graphs of the lamplighter group.
method Careful study of spectral properties of a one-parametric family of convolution operators on the lamplighter group.
result The spectrum of the discrete Laplacian on the Cayley graph of the lamplighter group is a union of an interval and a countable set of isolated points.
In this paper we introduce the area of 2-ruled 4-folds in R^n (n=7 or 8), that is, submanifolds M of R^n that admit a fibration over some 2-fold Sigma such that each fibre is an affine 2-plane in R^n. This is motivated by the paper math.DG/0012060 by Joyce on ruled special Lagrangian 3-folds in C^3 and the work of the …
We construct a Kaehler structure on the punctured cotangent bundle of the Cayley projective plane whose Kaehler form coincides with the natural symplectic form on the cotangent bundle and we show that the geodesic flow action is holomorphic and is expressed in a quite explicit form. We also give an embedding of the pun…
Upper bounds for Steklov eigenvalues in subgraphs of polynomial growth Cayley graphs.
problem Finding upper bounds for Steklov eigenvalues in subgraphs of polynomial growth Cayley graphs.
method Discretizing a bounded domain and using comparison theorems.
result The $k^{\mbox{th}}$ eigenvalue tends to 0 proportionally to 1/∣B∣d−11. This paper summarizes closed-form relations for SE(3) maps and their derivatives.
problem Closed-form expressions for SE(3) maps and their derivatives are scattered in the literature.
method Summarizes and provides proofs for relevant closed-form relations of the exponential and Cayley map on SE(3).
result Provides an implicit generalized-alpha scheme for rigid/flexible multibody systems using the Cayley map.