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.
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.
Since the work of Henri Cartan finite dimensional Riemannian symmetric spaces are an important subject of mathematical interest. They are related in a natural way to semisimple Lie groups. In this work we introduce and study their infinite dimensional generalization: Affine Kac-Moody symmetric spaces. Affine Kac-Moody …
New knots found with same determinant but no symmetric relation.
problem Determining if knots with the same determinant are symmetrically related.
method Constructing a family of knots with the same determinant but no symmetric relation.
result No two knots in the family are symmetrically related.
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.
New method reduces parameters for symmetric/antisymmetric relations in KBC.
problem Increased parameters for symmetric/antisymmetric relations in embedding-based KBC methods.
method L1 regularizer for Complex Embeddings to promote symmetry/antisymmetry.
result The proposed method outperforms baseline methods on the FB15k dataset.
Study provides explicit formula for complex 2D Kähler manifold quantization.
problem Quantization of complex 2D locally symmetric Kähler manifolds.
method Deformation quantization with separation of variables, solving recurrence relations.
result Explicit formula for star product on complex 2D locally symmetric Kähler manifolds.
Two related constructions are studied: (1) The diagonal complex D and its barycentric subdivision BD related to a \textit{punctured} oriented surface F equipped with a number of labeled marked points. (2) The symmetric diagonal complex Dinv and its barycentric subdivision $\math…
Defines Jacobi-Koszul-Vinberg structures on Jacobi-left-symmetric algebroids.
problem No specific problem stated; focuses on new structure definition.
method Definition and properties of Jacobi-Koszul-Vinberg structures on Jacobi-left-symmetric algebroids.
result Defines a new structure on Jacobi-left-symmetric algebroids.
Study on G2 actions on symmetric spaces, focusing on orbit properties.
problem Investigating properties of orbits in symmetric spaces related to G2. method Classification and analysis of orbits as Riemannian submanifolds, focusing on principal curvatures and specific types of orbits.
result Classification and properties of orbits in symmetric spaces related to G2. In this paper, we introduce the notion of a left-symmetric bialgebroid as a geometric generalization of a left-symmetric bialgebra and construct a left-symmetric bialgebroid from a pseudo-Hessian manifold. We also introduce the notion of a Manin triple for left-symmetric algebroids, which is equivalent to a left-symmet…
The oriented framed Homfly skein C of the annulus provides the natural parameter space for the Homfly satellite invariants of a knot. It contains a submodule C+ isomorphic to the algebra of the symmetric functions. We collect and expand formulae relating elements expressed in terms of symmetric functions to Turaev's ge…
Study of complexified Hermitian symmetric spaces and their structures.
problem Understanding the hyperkähler structures and symplectic properties of complexified Hermitian symmetric spaces.
method Explicit diffeomorphisms and moment-critical subsets analysis.
result Almost all complex and symplectic structures are equivalent to the ones on G/K and $T^*\left(G_u/K_0
ight)$ respectively. We systematically discuss connections on the spinor bundle of Cahen-Wallach symmetric spaces. A large class of these connections is closely connected to a quadratic relation on Clifford algebras. This relation in turn is associated to the symmetric linear map that defines the underlying space. We present various soluti…
We consider the problem of learning regression functions from pairwise data when there exists prior knowledge that the relation to be learned is symmetric or anti-symmetric. Such prior knowledge is commonly enforced by symmetrizing or anti-symmetrizing pairwise kernel functions. Through spectral analysis, we show that …
The article contains a survey of results on length-commensurable and isospectral locally symmetric spaces and related problems in the theory of semi-simple algebraic groups.
We introduce the notion of weak commensurabilty of arithmetic subgroups and relate it to the length equivalence and isospectrality of locally symmetric spaces. We prove many strong consequences of weak commensurabilty and derive from these many interesting results about isolength and isospectral locally symmetric space…
Geometrically connects Laplace eigenfunctions to Borel-Weil theory on symmetric spaces.
problem Understanding the spectral properties of Laplace-Beltrami operators on Riemannian symmetric spaces.
method Using symplectic geometry and geometric quantization, associating flag manifolds to symmetric spaces and relating their Satake diagrams.
result Harmonic polynomials on flag manifolds induce all eigenfunctions on symmetric spaces.
New exact sequence links cohomology, automorphisms, and extensions of symmetric quandles.
problem Understanding the structure of extensions and automorphisms in symmetric quandles.
method Derived a four-term exact sequence relating 1-cocycles, second cohomology, and automorphisms.
result Obstruction to automorphisms lies in the second cohomology of symmetric quandles.
We create a 3-skeleton for a symmetric group's classifying space.
problem Classifying space construction for symmetric groups.
method Combining rewriting systems and combinatorial methods.
result Correctness of the constructed 3-skeleton.
Study isotropy groups for complex orthogonal and skew-symmetric matrices.
problem Understanding isotropy subgroups of orthogonal similarity transformations.
method Analysis of group structure of nonsingular block matrices.
result Group structure of isotropy subgroups related to block Toeplitz matrices.
The study examines subgroups of braid groups related to symmetric groups.
problem Characterizing subgroups of braid groups that are extensions of symmetric groups.
method Analyzing normal subgroups of braid groups and their quotient structures.
result There are exactly 8 commensurability classes of such subgroups for n≥4. We present some basic results on a natural Poisson structure on any compact symmetric space. The symplectic leaves of this structure are related to the orbits of the corresponding real semisimple group on the complex flag manifold.
We provide an explicit example of a non trivial Legendrian knot Λ such that there exists a Lagrangian concordance from Λ0 to Λ where Λ0 is the trivial Legendrian knot. We then use the map induced in Legendrian contact homology by a concordance and the augmentation category of Λ to show that no Lagrangian co…
For a closed locally symmetric space M=Γ\G/K and a representation of G we consider the push-forward of the fundamental class in the homology of the linear group and a related invariant in algebraic K-theory. We discuss the nontriviality of this invariant and we generalize the construction to cusped locally symmetric sp…
The paper studies para-Sasaki-like manifolds with a new metric connection.
problem Investigating new geometric structures on para-Sasaki-like manifolds.
method Deriving relations between connections, analyzing curvature tensors, studying solitons, constructing examples.
result Derived relations and properties of para-Sasaki-like manifolds with the generalized symmetric metric connection.
There are two different approaches to exhibit submaximal symmetric rank 2 distributions in 5D via Monge equations. In this note we establish precise relations between these models, find auto-equivalences of one family, and treat two special equations.
The n-dimensional Lorentzian manifolds with vanishing second covariant derivative of the Riemann tensor (2-symmetric spacetimes) are characterized and classified. The main result is that either they are locally symmetric or they have a covariantly constant null vector field, in this case defining a subfamily of Brinkma…
The study proves a geometric result related to Harish-Chandra's theorem.
problem Understanding the relationship between symmetric submanifolds and Harish-Chandra's theorem.
method Analyzing maximal tori in Clifford tori within Euclidean spaces.
result A compact, intrinsically symmetric submanifold is extrinsically symmetric if and only if its maximal tori are Clifford tori.
We examine the moduli spaces of Type~A connections on oriented and unoriented surfaces both with and without torsion in relation to the signature of the associated symmetric Ricci tensor. If the signature of the symmetric Ricci tensor is (1,1) or (0,2), the moduli spaces are smooth. If the signature is (2,0), there is …
In this paper we prove mixed norm estimates for Riesz transforms related to Laplace--Beltrami operators on compact Riemannian symmetric spaces of rank one. These operators are closely related to the Riesz transforms for Jacobi polynomials expansions. The key point is to obtain sharp estimates for the kernel of the Jaco…
Classifies surfaces with special curvature properties.
problem Rotational surfaces with specific curvature conditions.
method Classifies surfaces with rotationally symmetric norms and linear curvature relations.
result Rotational surfaces with linearly related curvatures are classified.
The paper studies complex genera and related geometric applications, deriving formulas for multiple zeta values.
problem Understanding coefficients in Chern numbers for complex genera.
method Examining Chern numbers for complex genera, focusing on specific genera like Td^(1/2), Γ, and Todd.
result Unified formulas for multiple zeta values and transition matrices among symmetric functions.
Let M be an irreducible Riemannian symmetric space. The index i(M) of M is the minimal codimension of a totally geodesic submanifold of M. In previous work the authors proved that i(M) is bounded from below by the rank rk(M) of M. In this paper we classify all irreducible Riemannian symmetric spaces M for which the equ…
Study on Selberg's modified metric in symmetric spaces.
problem Properties of modified metric in symmetric spaces.
method Analysis of SL(n,R)/SO(n,R) with Selberg's premetric. result Generalizations of hyperbolic space properties.
Study cohomology rings of Grassmannians using Clifford algebras and symmetric spaces.
problem Understanding cohomology rings of Grassmannians over different fields.
method Explicit generators and relations for de Rham cohomology rings, filtered deformations related to Clifford algebras.
result Explicit generators and relations for the de Rham cohomology rings of Grassmannians.
In this article, relations between the root space decomposition of a Riemannian symmetric space of compact type and the root space decompositions of its totally geodesic submanifolds (symmetric subspaces) are described. These relations provide an approach to the classification of totally geodesic submanifolds in Rieman…
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.
Harmonic morphisms and p-harmonic functions constructed on symmetric spaces.
problem Constructing harmonic morphisms and p-harmonic functions on symmetric spaces.
method Using Cartan embedding and related maps to relate tension field and conformality operator.
result Simple formulae relating tension field and conformality operator on symmetric spaces to those on their images.
The paper studies quarter-symmetric connections on Hermitian and Kähler manifolds.
problem Examining quarter-symmetric connections on almost Hermitian and Kähler manifolds.
method Analyzing the curvature tensors and their properties with respect to quarter-symmetric connections.
result Constructed tensors that do not depend on the quarter-symmetric connection generator, including the Weyl projective curvature tensor.
The paper studies a new connection on Riemannian manifolds and finds conditions for symplectic manifolds.
problem Exploring a new quarter-symmetric non-metric connection on Riemannian manifolds.
method Analyzes the properties and relations of the torsion tensor and curvature tensors of the new connection.
result Conditions for a manifold to be symplectic when endowed with the new connection.
The paper connects Brauer algebra homology to symmetric group homology.
problem Understanding the homology of Brauer algebras.
method Interpreting Brauer algebras as Tor-groups and comparing to symmetric group homology.
result Isomorphism of homology groups under specific conditions.
Studying the isotropy orbits of compact symmetric spaces Reiswich introduced a family of explicit polynomials in one variable in order to describe the unique minimal isotropy orbit of compact symmetric spaces with Dynkin diagram of type Dm. Based on this geometric interpretation he conjectured that these polynomials…
Symmetrizes 4d and 3d BPS quivers for Argyres-Douglas theories.
problem Understanding the relationship between 4d and 3d BPS quivers.
method Analyzes geometric backgrounds and uses skein modules to derive quiver partition functions.
result Proves isomorphism between 4d wall-crossing and unlinking of symmetric quivers.
MIM learns useful representations with high mutual information.
problem Learning useful representations for downstream tasks.
method Symmetric Jensen-Shannon divergence and mutual information regularizer in an encoder/decoder framework.
result MIM learns high mutual information representations without posterior collapse.
New connections on symmetric spaces with invariant properties.
problem Understanding invariant connections on hermitian symmetric spaces.
method Introduced a class of G-invariant connections on homogeneous bundles over hermitian symmetric spaces. result Parameter space of connections is a normal variety with a canonical anti-holomorphic involution.
Defines extended TQFTs using handle attachments.
problem Constructing extended topological quantum field theories (TQFTs).
method Finite presentation of cobordism symmetric monoidal bicategory using handle attachments and relations.
result Constructs a once extended TQFT from categorified TQFT and handle 2-morphisms.
Explains rolling of symmetric spaces on flat spaces.
problem Clarifying the difference between two types of rolling.
method Detailed explanation and illustrative examples.
result Theoretical results complemented with examples.