The paper classifies symmetric triads with multiplicities and their applications.
problem Classifying symmetric triads with multiplicities and their applications.
method Developed the theory of symmetric triads with multiplicities, classified abstract triads, and determined corresponding triads for commutative compact triads.
result Classified symmetric triads with multiplicities and their applications.
Paper classifies compact symmetric triads using double Satake diagrams and canonical forms.
problem Classifying compact symmetric triads.
method Introducing double Satake diagrams and canonical forms, proving their existence and properties.
result Existence and properties of canonical forms for compact simple symmetric triads.
The paper establishes a duality between non-compact and compact symmetric pairs.
problem Understanding the relationship between non-compact and compact symmetric pairs.
method Developed a duality theorem between non-compact pseudo-Riemannian semisimple symmetric pairs and commutative compact semisimple symmetric triads.
result Explicit description of a one-to-one correspondence between non-compact and compact symmetric pairs.
We give a necessary and sufficient condition for orbits of commutative Hermann actions and actions of the direct product of two symmetric subgroups on compact Lie groups to be biharmonic in terms of symmetric triad with multiplicities. By this criterion, we determine all the proper biharmonic submanifolds in irreducibl…
Condition for intersection of real flag manifolds in complex flag manifold.
problem Intersection conditions of real flag manifolds in a complex flag manifold.
method Condition given in terms of symmetric triad, antipodal intersection proven.
result Intersection of real flag manifolds is antipodal.
We define the twisted Blanchfield pairing of a symmetric triad of chain complexes over a group ring Z[G], together with a unitary representation of G over an Ore domain with involution. We prove that the pairing is sesquilinear, and we prove that it is hermitian and nonsingular under certain extra conditions. A twisted…
Simplified construction of contact triad connection for analysis.
problem Analyzing contact instanton equation on contact triads.
method Characterization via almost contact structure and construction using almost contact moving frame.
result Simpler and more canonical construction of contact triad connection.
Homotopy theory for (2n+1)-dimensional manifold triads with fixed boundary.
problem Classifying stable moduli spaces of (2n+1)-dimensional manifold triads. method Homotopy-theoretic description of stable moduli spaces, stabilization by boundary connected sum with SnimesDn+1. result Established homology of stable moduli spaces for (2n+1)-dimensional manifold triads. We introduce a canonical affine connection on the contact manifold (Q,ξ), which is associated to each contact triad (Q,λ,J) where λ is a contact form and J:ξ→ξ is an endomorphism with J2=−id compatible to dλ. We call it the \emph{contact triad connection} of (Q,λ,J) and prove its existence and uniqu…
The (variational) graph auto-encoder and its variants have been popularly used for representation learning on graph-structured data. While the encoder is often a powerful graph convolutional network, the decoder reconstructs the graph structure by only considering two nodes at a time, thus ignoring possible interaction…
We consider differentiable maps in the setting of Abstract Differential Geometry and we study the conditions that ensure the uniqueness of differentials in this setting. In particular, we prove that smooth maps between smooth manifolds admit a unique differential, coinciding with the usual one. Thus smooth manifolds fo…
Generates sutured manifolds from a surface using surgery.
problem Creating sutured manifolds from a given surface.
method Using surgery triads to generate sutured manifolds.
result The set of sutured manifolds is generated by a specific manifold.
TSSC images enhance chaotic signal classification using ConvNets.
problem Classifying chaotic signals accurately and robustly.
method Triad State Space Construction (TSSC) for image encoding, Convolutional Neural Network (ConvNet) for classification.
result TSSC-ConvNet achieves high accuracy and robustness in chaotic signal classification.
Analyzes perturbed contact instantons with Legendrian boundary conditions using geometric analysis.
problem Analyzing nonlinear elliptic systems associated with contact Hamiltonian trajectories.
method Identifies correct action and energy functionals, develops elliptic regularity theory, and proves asymptotic convergence.
result Established C∞ convergence of perturbed contact instantons under finite energy hypothesis. New universal automorphic functions capture monstrous moonshine.
problem Developing a universal framework for automorphic functions.
method Reformulating old results, constructing new coordinates, and defining central extensions.
result New invariant 1-forms and representations for universal Teichmüller space.
Geodesic completeness proven for all compact locally symmetric Lorentz manifolds.
problem Geodesic completeness of compact locally symmetric Lorentz manifolds.
method Proof in all remaining cases using completeness result.
result All compact, locally symmetric Lorentz manifolds are geodesically complete.
Characterizes invariant contact metric structures on tangent sphere bundles of compact symmetric spaces.
problem Identifying invariant contact metric structures on tangent sphere bundles of compact symmetric spaces.
method Characterization through invariant Riemannian metrics and Killing vector fields.
result A family of invariant contact metric structures is obtained on tangent sphere bundles of compact symmetric spaces with rank greater than or equal to two.
Compact holonomy groups found in symmetric spaces.
problem Characterizing holonomy groups in symmetric spaces.
method Analyzing the holonomy group structure of locally symmetric spaces.
result Holonomy groups are compact and have finite index in the orthogonal group.
Proves stability of Einstein metrics on specific symmetric spaces.
problem Stability of Einstein metrics on symmetric spaces of compact type.
method Linear stability analysis of the Einstein-Hilbert action.
result Resolves stability problem for irreducible symmetric spaces of compact type.
Counting geodesics on compact symmetric spaces using orbit dimensions and topological data.
problem Counting geodesics on compact symmetric spaces.
method Using orbit dimensions and topological data of the symmetric space.
result Obtained data on dimensions and connected components of focal orbits.
Paper constructs an example of a non-compact submanifold in a quaternionic Kähler symmetric space.
problem Tackles the construction of a non-compact totally complex submanifold in a quaternionic Kähler symmetric space.
method Uses an isometric action of a compact Lie group and a maximal totally geodesic sphere.
result Proves the existence of a non-compact totally complex submanifold of maximal dimension in a compact quaternionic Kähler symmetric space.
We prove that a polar foliation of codimension at least three in an irreducible compact symmetric space is hyperpolar, unless the symmetric space has rank one. For reducible symmetric spaces of compact type, we derive decomposition results for polar foliations.
New compact minimal submanifolds found in Riemannian symmetric spaces.
problem Finding compact minimal submanifolds in Riemannian symmetric spaces.
method Constructing multi-dimensional families of compact minimal submanifolds via complex-valued eigenfunctions.
result New families of compact minimal submanifolds of codimension two in SU(n)/SO(n), Sp(n)/U(n), SO(2n)/U(n), and SU(2n)/Sp(n). New spectral analysis on non-compact spaces.
problem Analyzing pseudo-Riemannian locally symmetric spaces.
method Initiating spectral analysis beyond classical settings.
result Recent results in non-compact spaces.
New tools compute index of symmetry in homogeneous fibrations.
problem Computing the index of symmetry in homogeneous fibrations.
method Developed new tools to compute the index of symmetry.
result Determined the index of symmetry of various homogeneous spaces.
Paper proves geodesic ball maximizes second Robin eigenvalue in non-compact symmetric spaces.
problem Maximizing the second Robin eigenvalue in non-compact rank-1 symmetric spaces.
method Quantitative spectral inequality for the second Robin eigenvalue.
result Geodesic ball maximizes the second Robin eigenvalue among domains of the same volume.
Let Σ be a C3 compact symmetric convex hypersurface in R8. For some special cases, we prove that when Σ carries exactly four geometrically distinct closed characteristics, then all of them must be symmetric.
New theorems on compactness and finiteness for specific types of self-shrinkers.
problem Characterizing rotationally symmetric self-shrinkers with constraints.
method Compactness and finiteness theorems for self-shrinkers with specific symmetries and constraints.
result Existence of entropy minimizing self-shrinkers diffeomorphic to S1imesSn−1 for each n≥2. We prove that an isometric action of a compact Lie group on a compact symmetric space is variationally complete if and only if it is hyperpolar.
Study of a 32D Rosenfeld projective plane, a symmetric space.
problem Understanding the properties of a specific symmetric space.
method Detailed study and analysis of the geometric structure.
result Comprehensive understanding of the 32D Rosenfeld projective plane.
We give an explicit classification of maximal antipodal sets in any irreducible compact symmetric space except for spin groups and half spin groups, and some quotient symmetric spaces associated to them.
In two previous papers, we started a study of the first eigenvalue of the Dirac operator on compact spin symmetric spaces, providing, for symmetric spaces of "inner" type, a formula giving this first eigenvalue in terms of the algebraic data of the groups involved. We conclude here that study by giving the explicit exp…
New examples of isoparametric families on non-compact symmetric spaces.
problem Constructing isoparametric families with non-austere focal sets.
method New extension method from Euclidean spaces to symmetric spaces of non-compact type.
result First examples of isoparametric families on non-compact symmetric spaces.
New infinite families of flat spaces found from symmetric spaces.
problem Finding new flat homogeneous spaces.
method Starting from compact symmetric spaces, constructing infinite families of compact homogeneous spaces with invariant Bismut connections.
result Infinite families of compact homogeneous spaces with vanishing Ricci tensor.
Study on deformations of symmetric spaces using Jordan algebras.
problem Deformability of symmetric Einstein metrics on compact Lie algebras.
method Developed sandwich operators and quadratic Casimir operators for compact Lie algebras; calculated obstruction integrals from invariant polynomials; explored relation to simple Jordan algebras.
result Proved the nonlinear instability of most infinitesimally deformable irreducible compact symmetric spaces.
Study string topology on symmetric spaces, showing non-triviality and nilpotency results.
problem Understanding the structure of string topology on symmetric spaces.
method Used cycles from Bott-Samelson and Ziller to study coproduct and product.
result Showed non-triviality and nilpotency of Chas-Sullivan product and coproduct for higher rank symmetric spaces.
Volume comparison theorem for rank 1 symmetric spaces proved.
problem Volume comparison for symmetric spaces of non-compact type.
method Normalized Ricci--DeTurck flow to analyze volume functional and derive monotonicity properties.
result Volume comparison theorem established for rank 1 symmetric spaces of non-compact type.
Paper finds obstructions to compact Clifford-Klein forms for tangential symmetric spaces.
problem Existence of compact Clifford-Klein forms for tangential symmetric spaces.
method Analyzes tangential homogeneous spaces and provides necessary conditions for their existence.
result New tangential symmetric spaces are found that do not admit compact Clifford-Klein forms.
New Einstein manifolds split into symmetric and compact parts.
problem Understanding Einstein manifolds with unimodular isometry groups.
method Theory of polar actions, Lie-theoretic arguments, and maximum principles.
result Negative Einstein manifolds split into symmetric and compact parts.
We prove that the orbits of a polar action of a compact Lie group on a compact rank one symmetric space are tautly embedded with respect to Z_2-coefficients.
We study hyperpolar actions on reducible symmetric spaces of the compact type. Our main result is that an indecomposable hyperpolar action on a symmetric space of the compact type is orbit equivalent to a Hermann action or of cohomogeneity one.
In this paper, we obtain a Cartan type identity for curvature-adapted isoparametric hypersurfaces in symmetric spaces of compact type or non-compact type. This identity is a generalization of Cartan-D'Atri's identity for curvature-adapted(=amenable) isoparametric hypersurfaces in rank one symmetric spaces. Furthermore,…
Near isospectrality forces full isospectrality for compact quotients of symmetric spaces.
problem Inverse spectral problem for Riemannian manifolds
method Proving near isospectrality implies full isospectrality
result Compact quotients of symmetric spaces have full isospectrality
Harmonic functions on compact symmetric spaces exhibit strong convexity properties.
problem Understanding the convexity of harmonic functions on compact symmetric spaces.
method Analyzing the nonnegativity of the Laplacian powers of harmonic functions.
result Harmonic functions on compact symmetric spaces have nonnegative Laplacian powers, demonstrating strong convexity.
Method calculates spectra of Rarita-Schwinger operator on symmetric spaces.
problem Calculating spectra of the Rarita-Schwinger operator on compact symmetric spaces.
method Using Weitzenböck formulas, Laplace operator, Casimir operator, Freudenthal's formula, and branching rules.
result Obtained spectra on the sphere, complex projective space, and quaternionic projective space.
We obtain the full classification of coisotropic and polar actions of compact Lie group on irreducible Hermitian symmetric spaces.
Paper proves eigenvalue inequality for Hopf-symmetric domains.
problem Eigenvalue inequality for Hopf-symmetric domains in non-compact symmetric spaces.
method Used geometric and spectral analysis on non-compact rank one symmetric spaces.
result Eigenvalue inequality for bounded Hopf-symmetric domains in non-compact symmetric spaces.
Compact rank one symmetric spaces are rigid under certain curvature conditions.
problem Rigidity of compact rank one symmetric spaces under curvature constraints.
method Examined compact symmetric spaces with metric g0 of rank one, and another metric g with sectional curvature bounded by 0 to 1. result If g equals g0 outside a convex subset, then g is isometric with g0.