The study restricts Anosov subgroups of Sp(2n,R) based on subset Θ.
problem Characterizing Anosov subgroups of Sp(2n,R) based on subset Θ.
method Analyzing the structure of Anosov subgroups in terms of subset Θ.
result Anosov subgroups of Sp(2n,R) are virtually free or surface groups if Θ contains an odd integer, otherwise they are not.
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.
Survey on geometry and topology of maximal antipodal sets.
problem Maximal antipodal sets on Riemannian manifolds.
method Comprehensive survey of existing research.
result Relation to various mathematical areas.
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.
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. 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.
We classify R-spaces that admit a certain natural Γ-symmetric structure. We further determine the maximal antipodal sets of these structures.
Given any finite subset X of the sphere S^n, n>1, which includes no pairs of antipodal points, we explicitly construct smoothly immersed closed orientable hypersurfaces in Euclidean space R^{n+1} whose Gauss map misses X. In particular, this answers a question of M. Gromov.
The paper classifies 10 antipodal pairings of self-dual maps.
problem Understanding the antipodal pairings of strongly involutive polyhedra.
method Classification of self-dual pairings and construction of polyhedra.
result Determination of 10 antipodal pairings among 24 self-dual pairings.
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.
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.
Spheres can be stretched to have larger diameter than antipodal distance.
problem Finding the maximum diameter of spheres relative to antipodal points.
method Deformation of spheres in dimensions ≥3.
result Spheres can be deformed to have diameter larger than antipodal distance.
Study finds a minimum volume for vector fields on a punctured sphere.
problem Finding the minimum volume of unit vector fields on a punctured sphere.
method Analyzes the volume of vector fields tangent to an antipodally punctured unit 2-sphere.
result Provides a lower bound for the volume of unit vector fields.
Invariant structures link to algebraic curves with specific properties.
problem Linking invariant hypercomplex structures to algebraic curves.
method Mapping invariant structures to algebraic curves with specific properties.
result Invariant hypercomplex structures correspond to algebraic curves with a flat projection and antiholomorphic involution.
Proves Poincaré duality for Hopf algebroids with bijective antipode.
problem Proving Poincaré duality for Hopf algebroids.
method Using twisted Poincaré duality and bijective antipode properties.
result Recovering and extending known Poincaré dualities for Hopf algebroids.
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…
The aim of this paper is a characterization of great antipodal sets of complex Grassmannian manifolds as certain designs with the smallest cardinalities.
We classify the volume preserving stable hypersurfaces in the real projective space RPn. As a consequence, the solutions of the isoperimetric problem are tubular neighborhoods of projective subspaces RPk⊂RPn (starting with points). This confirms a conjecture of Burago and Zalgal…
The classical Lusternik-Schnirelman-Borsuk theorem states that if a d-sphere is covered by d+1 closed sets, then at least one of the sets must contain a pair of antipodal points. In this paper, we prove a combinatorial version of this theorem for hypercubes. It is not hard to show that for any cover of the facets of a …
The study uses symplectic capacities to bound the systole on the sphere.
problem Bounding the systole on the sphere using symplectic capacities.
method Using symplectic capacities and properties of fiberwise balanced hypersurfaces.
result Upper bounds on the systole in terms of geometric data and β. Paper defines conditions for projective links in projective 3-space.
problem Characterizing links in projective 3-space.
method Combinatorial conditions and antipodal symmetry.
result Easy condition to prevent alternating projective links.
Study sharp upper bounds for Aharonov-Bohm eigenvalues on surfaces.
problem Finding sharp upper bounds for eigenvalues of magnetic Laplacian.
method Isoperimetric inequalities and bounds in terms of Gaussian curvature.
result Maximal first eigenvalue for geodesic disk on simply connected surfaces.
The paper explores symmetric representations of links and conditions for amphichirality.
problem Investigating symmetric representations of links and conditions for amphichirality.
method Using antipodally self-dual and antipodally symmetric maps, the authors provide sufficient combinatorial conditions for amphichirality.
result A link is amphichiral if its self-dual pairing is not one of 6 specific ones.
The paper proves the existence of minimal surfaces avoiding specific points.
problem Proving the existence of minimal surfaces avoiding specific points.
method Interpolation theorem for conformal minimal immersions avoiding hyperplanes.
result Existence of complete conformal minimal immersions avoiding prescribed points.
We show that the volume of any Riemannian metric on a three sphere is bounded below by the length of the shortest closed curve that links its antipodal image. In particular, the volume is bounded below by the minimum of the length of the shortest closed geodesic and the minimal distance between antipodal points.
The total diameter of a closed planar curve C⊂R2 is the integral of its antipodal chord lengths. We show that this quantity is bounded below by twice the area of C. Furthermore, when C is convex or centrally symmetric, the lower bound is twice as large. Both inequalities are sharp and the equality holds i…
For p∈(1,2] and a bounded, convex, nonempty, open set Ω⊂R2 let μp(Ωˉ,⋅) be the p-capacitary curvature measure (generated by the closure Ωˉ of Ω) on the unit circle S1. This paper shows that such a problem of prescribing μp on a planar convex domain: "Given a finite…
Grafting a measured lamination on a hyperbolic surface defines a self-map of Teichmuller space, which is a homeomorphism by a result of Scannell and Wolf. In this paper we study the large-scale behavior of pruning, which is the inverse of grafting. Specifically, for each conformal structure $X \in \T(S)$, pruning X g…
Minimal vector fields on a 2-sphere with varying volumes are discovered.
problem Minimal vector fields on a 2-sphere with specific properties.
method Homology theory of the unit tangent bundle, calibrations, and minimal volume equation.
result A family of minimal vector fields with unbounded volume and another with smaller volume than known optimal fields.
Minimal surfaces in S3(2) linked to vector fields on punctured sphere.
problem Connecting minimal surfaces in S3(2) to vector fields on a punctured sphere.
method Established a correspondence between minimal surfaces and area-minimizing vector fields.
result Stability relation for Lawson cylinders in S3(2).
Flat subsets in Euclidean buildings are contained within apartments.
problem Understanding the structure of flat subsets in Euclidean buildings.
method Proving containment within apartments.
result Convex flat subsets are contained in apartments.
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.
Study finds the volume of unit vector fields on a punctured sphere and shows their images match minimally immersed Klein bottles.
problem Finding the volume of unit vector fields on a punctured sphere.
method Analyzes the volume of unit vector fields on an antipodally punctured unit 2-sphere and shows their images coincide with minimally immersed Klein bottles.
result The images of minimizing vector fields on the punctured sphere match those of minimally immersed Klein bottles.
Study on Borel Anosov subgroups in SL(d,R) for d≠5,8k±1.
problem Characterizing Borel Anosov subgroups in SL(d,R).
method Analysis of antipodal subsets and quasi-isometric embeddings.
result Borel Anosov subgroups are virtually free or hyperbolic surface groups.
Algorithm improves recommendation subset selection in the presence of biases.
problem Maximizing submodular functions for recommendation in the presence of social biases.
method Algorithm for submodular maximization with fairness constraints.
result Algorithm provably outputs subsets with near-optimal utility and proportional representation.
Study on embeddings and their topological properties in R^d.
problem Topology of embeddings and their bounds in R^d.
method Combinatorial formula for upper bounds of embeddings into R^d.
result Simple combinatorial formula for upper bounds of embeddings into R^d.
We characterize the three-dimensional spaces admitting at least six or at least seven equidistant points. In particular, we show the existence of C∞ norms on R3 admitting six equidistant points, which refutes a conjecture of Lawlor and Morgan (1994, Pacific J. Math \textbf{166}, 55--83), and gives the exist…
We introduce and study a new class of representations of surface groups into Lie groups of Hermitian type, called {\em weakly maximal} representations. We prove that weakly maximal representations are discrete and injective and we describe the structure of the Zariski closure of their image. Furthermore we prove that t…
Study dihedral spherical surfaces and their foliations.
problem Characterize dihedral spherical surfaces and their foliations.
method Define and analyze dihedral surfaces and their foliations, introduce geometric decompositions and deformations.
result Determine the dimension of the moduli space for dihedral surfaces.
In these short notes we characterize the loxodromic unit vector fields on antipodally punctured Euclidean spheres as the only ones achieving a lower bound for the volume functional depending on the Poincaré indexes around their singularities.
We give examples of Lie-Rinehart algebras whose enveloping algebra is not a full Hopf algebroid in the sense of Bohm and Szlachanyi. We construct these examples as quotients of a canonical Lie-Rinehart algebra over a Jacobi algebra which does admit an antipode.
The paper creates a deformation retraction for homeomorphisms of the projective plane.
problem Deformation retraction of homeomorphisms of the projective plane.
method Equivariant strong deformation retraction from homeomorphism group to special orthogonal group.
result Induces a SO(3)-equivariant strong deformation retraction from projective plane homeomorphisms to SO(3).
The paper reduces normal curvature and enhances homology recovery via embedded submanifolds.
problem Recovering the homology of submanifolds with narrow cycles.
method Embedding submanifolds into scaled oriented Grassmannian bundles to reduce normal curvature and stabilize Čech persistent homology.
result The Čech persistent homology is stable with respect to the interleaving distance and provides lower bounds on scales for homology recovery.
In this paper we study singular points of the Wigner caustic and affine λ--equidistants of planar curves based on shapes of these curves. We generalize the Blaschke-Süss theorem on the existence of antipodal pairs of a convex curve.
In the present paper we study the structure of the cut locus of a Randers rotational 2-sphere of revolution (M,F=α+β). We show that in the case when the Gaussian curvature of the Randers surface is monotone along a meridian, the cut locus of a point q∈M is a point on a subarc of the opposite half bending meri…
Designs efficient algorithms to maximize the expectation of Gaussian random variables.
problem Maximizing the expectation of the supremum of Gaussian random variables.
method Polynomial time approximation scheme and O(logn) approximation algorithm for general m>1. result Characterizes optimal variance allocation and provides approximation algorithms.
Stokes' theorem's boundary maximizes entropy.
problem Characterizing the boundary of a manifold using entropy.
method Maximizing entropy for codimension-1 submanifolds satisfying Stokes' theorem.
result The boundary of a manifold maximizes the entropy functional.
Biholomorphisms of transport twistor spaces are rigid under certain conditions.
problem Rigidity of biholomorphisms in transport twistor spaces.
method Proof of rigidity for biholomorphisms between transport twistor spaces of simple or Anosov surfaces.
result Biholomorphisms are rigid, up to constant rescaling and the antipodal map, being lifts of orientation-preserving isometries.