Study on totally symmetric sets with group applications.
problem Understanding totally symmetric sets and their group applications.
method Survey of existing theory and applications to various groups.
result Exploration of totally symmetric sets in multiple group contexts.
Study maximal antipodal sets in exceptional symmetric spaces.
problem Classify maximal antipodal sets in exceptional symmetric spaces.
method Combining existing literature and new results, classify maximal antipodal sets.
result Complete classification of maximal antipodal sets in all exceptional compact symmetric spaces.
Proves conjectures about maximal antipodal sets in symmetric and generalised symmetric spaces.
problem Cohomological descriptions of maximal antipodal sets in symmetric spaces.
method Equivariant cohomology theory.
result Proves several long-standing conjectures by Chen--Nagano and extends them to generalised symmetric spaces.
The study classifies and characterizes totally symmetric sets in the general linear group.
problem Understanding the structure and properties of totally symmetric sets in the general linear group.
method Formulated a notion of irreducibility for totally symmetric sets in the general linear group and classified them.
result Classification of irreducible totally symmetric sets and those of maximal cardinality.
Study shows non-symmetric convex sets have full boundary limits.
problem Understanding boundaries of non-symmetric convex sets.
method Proved using proximal limit set analysis.
result Proximal limit set equals full projective boundary for non-symmetric irreducible divisible convex sets.
Paper classifies totally symmetric sets in groups and bounds their sizes.
problem Understanding homomorphisms between groups using totally symmetric sets.
method Full classifications and size bounds for totally symmetric sets in various groups.
result Derives restrictions on homomorphisms between certain groups.
Characterizes higher rank model geometries using antipodal sets.
problem Identifying higher rank model geometries among Hadamard spaces.
method Using antipodal sets at infinity to characterize model geometries.
result Characterizes Riemannian symmetric spaces, Euclidean buildings, and products as higher rank model geometries.
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.
Researchers found a maximal antipodal set of three elements in a 7x7 sphere space.
problem Determining the maximal antipodal set in the outer 3-symmetric space S7imesS7. method Investigated the polar and maximal antipodal set P for the given 3-symmetric space S7imesS7. result The maximal antipodal set P has three elements. Study continuity of limit sets in symmetric spaces.
problem Continuity of limit sets for geometrically finite subgroups in symmetric spaces.
method Extended geometrically finite representations theory.
result Limit sets vary continuously with respect to Hausdorff distance under strong convergence.
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.
The paper defines generalized s-manifolds and explores their polars and antipodal sets.
problem Understanding polars and antipodal sets in generalized s-manifolds.
method Introduced generalized s-manifolds and provided a method to construct them. Studied polars and antipodal sets.
result Extended results on compact symmetric spaces to generalized s-manifolds.
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.
In this paper, a correspondence via duality is established between the set of locally strongly convex symmetric equiaffine hyperspheres and the set of minimal symmetric Lagrangian submanifolds in a certain complex space form. By using this correspondence theorem, we are able to provide an alternative proof of the class…
Simplified computation of symmetric gl_1 homology for links.
problem Computing the symmetric gl_1 homology for uncolored links.
method Down-to-earth description, basis construction, and algorithm for computation.
result An algorithm and program for computing the invariant for uncolored links.
Study proposes initial data sets for solving gravitational equations, proving energy estimates.
problem Solving the constraint equations in the evolutionary form.
method Proposes a family of initial data sets, proving Penrose-like energy estimates.
result Established existence of solutions for specific cases.
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.
The paper proves that symmetric sets with minimal Gaussian surface area are nearly convex cylinders.
problem Finding the shape of symmetric sets with minimal Gaussian surface area.
method Analyzing the boundary of symmetric sets and applying isoperimetric inequalities.
result Symmetric sets with minimal Gaussian surface area are nearly convex cylinders.
Paper proves new inequalities for Einstein-Maxwell data sets.
problem Establishing area-charge inequalities for Einstein-Maxwell initial data sets.
method Applying Gromov's μ-bubble technique in a new geometric context.
result Novel rigidity theorems for noncompact Einstein-Maxwell data sets.
We construct almost complex algebraic curvature tensors for pseudo Hermitian inner products whose skew-symmetric curvature operator has constant Jordan normal form on the set of non-degenerate complex lines.
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.
Minimal submanifolds in matrix spaces proven for specific ranks.
problem Minimal submanifolds in matrix spaces.
method Proving semialgebraic sets of matrices are minimal.
result Rectangular, skew-symmetric, and symmetric matrices with prescribed eigenvalues are minimal.
We classify R-spaces that admit a certain natural Γ-symmetric structure. We further determine the maximal antipodal sets of these structures.
The main result implies that a proper convex subset of an irreducible higher rank symmetric space cannot have Zariski dense stabilizer.
Finite singular times for symmetric network curvature flow.
problem Formation of singularities in network curvature flow.
method Curvature flow of networks with symmetric initial data and two triple junctions.
result The set of singular times is finite.
We provide a finite dimensional categorification of the symmetric evaluation of slN-webs using foam technology. As an output we obtain a symmetric link homology theory categorifying the link invariant associated to symmetric powers of the standard representation of slN. In addition, the cons…
It is known that the antipodal set of a Riemannian symmetric space of compact type G/K consists of a union of K-orbits. We determine the dimensions of these K-orbits of most irreducible symmetric spaces of compact type. The symmetric spaces we are not going to deal with are those with restricted root system $\m…
We summarize recent results initiating spectral analysis on pseudo-Riemannian locally symmetric spaces Γ\G/H, beyond the classical setting where H is compact (e.g. theory of automorphic forms for arithmetic Γ) or Γ is trivial (e.g. Plancherel-type formula for semisimple symmetric spaces).
In this paper we show that if the limit set is not small ,marked length spectrum determines geometric structure of rank one locally symmetric manifolds.
The paper studies billiards in symmetric tables and finds a measure bound for maximizing orbits.
problem Understanding the measure of maximizing orbits in symmetric billiard tables.
method Introduced a closed invariant set of locally maximizing orbits and gave an effective bound on its measure.
result An effective bound on the measure of the invariant set in terms of the isoperimetric defect of the curve.
Proves conditions for Fourier transforms in rank 1 symmetric spaces.
problem Understanding Fourier transform bounds in symmetric spaces.
method Proves sufficient and necessary conditions using Lipschitz and Fourier type integral conditions.
result Establishes bounds for Fourier transforms in rank 1 symmetric spaces with specific moduli of continuity.
Symmetric TSP is structurally equivalent to a constrained Group Steiner Tree Problem.
problem Finding the shortest tour in a symmetric TSP.
method Structural equivalence between symmetric TSP and constrained Group Steiner Tree Problem.
result Maximizing net weight in the cGSTP is equivalent to minimizing the TSP tour length.
Lie PCA improves density estimation on symmetric manifolds.
problem Density estimation for symmetric manifolds.
method Spectral method to approximate Lie algebra of symmetry group.
result Improved sample complexity and density estimation on various data sets.
The paper proposes a conjecture for a symmetric version of Ehrhard's inequality.
problem Formulating a conjecture for the optimal Ehrhard-type inequality for convex symmetric sets.
method Formulating a conjecture and explaining its optimality in terms of Gaussian concavity power.
result Proving certain inequalities for symmetric convex sets, with round k-cylinders as the only equality cases.
Study special affine connections on symmetric spaces and their products.
problem Characterize special affine connections on symmetric spaces.
method Analyze canonical affine connections, introduce special products, and study holonomy Lie algebras.
result Established a correspondence between special affine connections and special products on the Lie algebra.
The non-existence of non-trivial conformally symmetric manifolds in the three-dimensional Riemannian setting is shown. In Lorentzian signature, a complete local classification is obtained. Furthermore, the isometry classes are examined.
In this paper, we provide a concrete interpretation of equivariant Reidemeister torsion and demonstrate that Bismut-Zhang's equivariant Cheeger-Müller theorem simplifies considerably when applied to locally symmetric spaces. In a companion paper, this allows us to extend recent results on torsion cohomology growth and …
Local equivalence found between maximally symmetric rolling and flat Cartan distributions.
problem Establishing local equivalence between maximally symmetric rolling and flat Cartan distributions.
method Using complex parametrisation of su(2), a change of coordinates maps the maximally symmetric rolling (2,3,5)-distribution to the flat Cartan distribution. result Local equivalence between maximally symmetric rolling and flat Cartan distributions established.
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
Having developed a description of indefinite extrinsic symmetric spaces by corresponding infinitesimal objects in the preceding paper we now study the classification problem for these algebraic objects. In most cases the transvection group of an indefinite extrinsic symmetric space is not semisimple, which makes the cl…
SACP aggregates nonconformity scores from multiple predictors to create more efficient uncertainty sets.
problem Combining predictive uncertainties from multiple models for efficient and reliable uncertainty quantification.
method SACP (Symmetric Aggregated Conformal Prediction) aggregates nonconformity scores using a flexible symmetric aggregation function.
result SACP consistently improves efficiency and often outperforms state-of-the-art model aggregation baselines.
Characterizes the local diffeomorphism structure of the exponential in the set of skew-symmetric matrices.
problem Characterizing the local diffeomorphism structure of the exponential in the set of skew-symmetric matrices.
method Introduce the diffeomorphic logarithm of special orthogonal matrices and an efficient algorithm.
result The region containing the principal logarithm has a special multiplicity structure.
Study differential properties of matrix square roots in specific cases.
problem Understanding matrix square roots in semi-simple, symmetric, and orthogonal cases.
method Analysis of differential and metric structures of real square roots of matrices under specific conditions.
result Differential properties of matrix square roots in semi-simple, symmetric, and orthogonal cases.
This work improves understanding of symmetrizing Bregman divergences on positive definite matrices.
problem Understanding which mean to use for symmetrizing Bregman divergences on positive definite matrices.
method Axiomatic definition of mean functionals and variational principles over the cone of positive definite matrices.
result The arithmetic mean is canonical for forward symmetrization, and the arithmetic, log-Euclidean, and harmonic means for reverse symmetrization.
We study quasi-isometric embeddings of symmetric spaces and non-uniform irreducible lattices in semisimple higher rank Lie groups. We show that any quasi-isometric embedding between symmetric spaces of the same rank can be decomposed into a product of quasi-isometric embeddings into irreducible symmetric spaces. We thu…
In his paper "Shapes of Polyhedra and Triangulations of the Sphere", Thurston found that the set of shapes of convex polyhedra with prescribed cone-deficits has a complex hyperbolic structure. Inspired by his work, this paper studies the set of shapes of centrally symmetric octahedra with prescribed cone-deficits. We s…
We show that every limit point of a Zariski dense discrete subgroup Γ of the isometry group of a symmetric space of noncompact type is conical if and only if Γ is convex cocompact.
We classify the connected pseudo-Riemannian manifolds of signature (p,q) with q≥5 so that at each point of M the skew-symmetric curvature operator has constant rank 2 and constant Jordan normal form on the set of spacelike 2 planes and so that the skew-symmetric curvature operator is not nilpotent for at least …