The paper bounds entries of γ-vectors for flag homology spheres.
problem Bounding entries of γ-vectors for flag homology spheres.
method Structural and enumerative results for flag homology spheres.
result Supports a conjecture and characterizes structures of flag homology spheres.
Given a Coxeter system (W,S), there is an associated CW-complex, Sigma, on which W acts properly and cocompactly. We prove that when the nerve L of (W,S) is a flag triangulation of the 3-sphere, then the reduced ℓ2-homology of Sigma vanishes in all but the middle dimension.
The based loop space homology of a special family of homogeneous spaces, flag manifolds of connected compact Lie groups is studied. First, the rational homology of the based loop space on a complete flag manifold is calculated together with its Pontrjagin structure. Second, it is shown that the integral homology of the…
The study classifies transverse spheres in flag manifolds and finds new examples.
problem Classifying transverse spheres in flag manifolds.
method Using topological K-theory and constructions of transverse spheres.
result Classification of transverse spheres in various flag manifolds.
A new axiom for Finsler geometry leads to constant flag curvature.
problem Understanding Finslerian manifolds and their properties.
method Proposing and proving an axiom of spheres.
result Finslerian manifolds satisfying the axiom of spheres have constant flag curvature.
New f-vectors reveal geometric Lefschetz-like decompositions of flag spheres.
problem Understanding f-vectors of balanced simplicial complexes and flag spheres. method Analyzing h-vectors and f-vectors of flag spheres and balanced simplicial complexes. result Found f-vectors leading to geometric Lefschetz-like decompositions. Classifies minimal immersions from S2 into specific flag manifolds.
problem Classifying minimal immersions from S2 into specific flag manifolds. method Classification based on constant curvature and low-dimensional flag manifolds.
result Primitive minimal immersions of constant curvature from S2 into F2,1,1 and F2,2,1 are classified. The paper computes the cellular homology of real flag manifolds.
problem Computing the cellular homology of real flag manifolds.
method Explicit parametrizations of Schubert cells by closed balls (cubes) in R^n, using them to compute the boundary operator.
result Explicit formula for the boundary operator with refined coefficients.
In this paper, we investigate a relation between finite graphs, simplicial flag complexes and right-angled Coxeter groups, and we provide a class of reconstructible finite graphs. We show that if Γ is a finite graph which is the 1-skeleton of some simplicial flag complex L which is a homology manifold of dimension …
Homogeneous Finsler spheres with constant curvature have specific geodesic properties.
problem Existence and properties of homogeneous Finsler spheres with constant flag curvature.
method Proofs and analysis of geodesic properties on homogeneous Finsler spheres.
result Homogeneous Finsler spheres with constant flag curvature are either Riemannian or Randers.
Study on flag manifolds using Hopf fibration to find geometrically special 2-spheres.
problem Computing and characterizing 2-spheres in flag manifolds.
method Hopf fibration, invariant geometry, Weyl group action.
result Generators of second homotopy group have invariant geometry and are classified.
Quaternionic Brownian motion on flag manifold linked to sphere diffusion.
problem Modeling quaternionic stochastic areas on quaternionic flag manifolds.
method Relating quaternionic Brownian motion to symplectic Brownian motion and using radial dynamics.
result Quaternionic stochastic areas follow a multivariate normal distribution.
The Pachner graph of 2-spheres is studied, focusing on subgraphs of flag and stacked 2-spheres.
problem Characterize subgraphs of the Pachner graph of 2-spheres.
method Analyzes various induced subgraphs of the Pachner graph of n-vertex triangulated 2-spheres, proving connectivity and providing bounds on the number of connected components. result The subgraph of n-vertex flag 2-spheres is connected, while the subgraph of n-vertex stacked 2-spheres has at least as many connected components as trees with specific properties. We construct a categorification of the maximal commutative subalgebra of the type A Hecke algebra. Specifically, we propose a monoidal functor from the (symmetric) monoidal category of coherent sheaves on the flag Hilbert scheme to the (non-symmetric) monoidal category of Soergel bimodules. The adjoint of this functo…
We use group homology to define invariants in algebraic K-theory and in an analogue of the Bloch group for Q-rank one lattices and for some other geometric structures. We also show that the Bloch invariants of CR structures and of flag structures can be recovered by a fundamental class construction.
The paper extends geodesic orbit sphere classification to Finsler geometry.
problem Classifying geodesic orbit spheres in Finsler geometry.
method Generalized from Riemannian to Finsler geometry, proving constant curvature conditions.
result Geodesic orbit Finsler spheres with constant flag curvature are Randers.
The study classifies discrete pseudomanifolds with up to 2d+7 vertices.
problem Understanding discrete pseudomanifolds with a small number of vertices.
method Proved existence of at least 2(d+1) vertices, classified up to 2d+6 vertices, established equivalence with edge graphs of flag normal pseudomanifolds.
result Every flag normal d-pseudomanifold with at most 2d+7 vertices is either a simplicial d-sphere or a flag triangulation of the (d-2)-fold suspension of RP^2.
Study the module structure of homology of Artin kernels.
problem Characterize the module structure of homology of Artin kernels.
method Use flag complex and double covers of toric complexes to analyze properties of torsion part.
result Determine dimensions and sizes of Jordan forms of the torsion part.
MuRiT efficiently computes multi-parameter persistence barcodes.
problem Efficient computation of multi-parameter persistent homology.
method Vietoris-Rips transformation to reduce multi-parameter to single-parameter computation.
result MuRiT computes pathwise persistence barcodes for multi-filtered flag complexes.
Knot homology of Coxeter links identified with line bundles on Hilbert schemes.
problem Identifying knot homology for Coxeter links.
method Using line bundles on a generalized flag Hilbert scheme of points in \(\mathbb{C}^2\).
result Knot homology of Coxeter links corresponds to sections of a specific line bundle.
Associated to any finite flag complex L there is a right-angled Coxeter group W_L and a contractible cubical complex Sigma_L (the Davis complex) on which W_L acts properly and cocompactly, and such that the link of each vertex is L. It follows that if L is a generalized homology sphere, then Sigma_L is a contractible h…
Study geodesics on spheres with constant curvature, showing integrability and invariant properties.
problem Characterizing geodesics on spheres with constant flag curvature.
method Analyzing non-reversible Finsler metrics on S2 with constant flag curvature 1. result Geodesic flow is conjugate to Katok's examples and length of shortest closed geodesic is invariant.
For each partial flag manifold of SU(N), we define a Floer homology theory for knots in 3-manifolds, using instantons with codimension-2 singularities.
The study shows how to construct d-spheres from (d−1)-spheres and d-balls without additional vertices.
problem Constructing d-spheres from (d−1)-spheres and d-balls without additional vertices. method Examining specific types of spheres (flag, stacked, join of spheres) and d-balls to determine if constructions can be made without extra vertices. result Affirmative answers to constructing d-spheres from (d−1)-spheres and d-balls without additional vertices for certain types of spheres and d-balls. Study on constant curvature immersions of surfaces into flag manifolds.
problem Investigate constant curvature immersions of Riemann surfaces into flag manifolds.
method Investigate pseudoholomorphic maps and invariant metrics on flag manifolds.
result Unitarily equivalent primitive immersions of the two-sphere into full flag manifolds have constant curvature under all invariant metrics.
The paper computes the de Rham cohomology of real flag manifolds.
problem Computing the second de Rham cohomology group of real flag manifolds.
method Using the Weil construction and computations of the second homology group.
result The second de Rham cohomology group is zero in general, with some exceptions.
Study isoparametric hypersurfaces in a Randers sphere with constant flag curvature.
problem Characterize isoparametric hypersurfaces in a Randers sphere of constant flag curvature.
method Analyze isoparametric hypersurfaces for the standard round sphere and show their behavior under navigation.
result Provide a classification of special isoparametric hypersurfaces and their ambient metrics.
The famous Banach-Tarski paradox claims that the three dimensional rotation group acts on the two dimensional sphere paradoxically. In this paper, we generalize their result to show that the classical group acts on the flag manifold paradoxically.
Holomorphic structures on quantum flag manifolds uniquely defined.
problem Defining unique holomorphic structures on quantum flag manifolds.
method Constructing covariant q-deformed holomorphic structures. result Holomorphic structures are unique for simple relative Hopf modules.
Two exceptional flag manifolds' complex structures are studied, proving rigidity for one.
problem Proving rigidity of complex structures on two exceptional flag manifolds.
method Homogeneous Kähler manifolds under G2 group, relating to sphere complex structures. result Rigidity of complex structure proved for one manifold.
The paper examines how edge subdivisions affect the vanishing of L2-homology in Coxeter groups.
problem The vanishing of L2-homology in Coxeter groups under edge subdivisions. method Investigates conditions for the vanishing of L2-homology to be preserved under edge subdivisions of flag triangulations. result Conditions are given to preserve the vanishing of L2-homology under edge subdivisions, and counterexamples are constructed for a torsion growth analogue of Singer's conjecture. New Finslerian Ressiner-Nordstrom spacetime with constant flag curvature.
problem Exploring a new spacetime model with constant flag curvature.
method Derived Finslerian Ressiner-Nordstrom solution and analyzed its properties.
result The solution differs from Ressiner-Nordstrom metric only in two dimensional subspace with constant flag curvature.
Method proves complex homeomorphic to a sphere using bisimplices.
problem Proving regular CW complexes homeomorphic to spheres.
method Discrete Morse theory and bisimplices.
result Flag bisimplicial completion of quadric complexes is contractible.
In this note we show that the configuration spaces of the kinematic system constructed in [4] and [12] gives rise to a natural tower of sphere bundles. Moreover, we prove that, each tower of projective bundles associated to special multi- flags (cf [1], [13], [2], [3]), we can associate such a tower of sphere bundles w…
New theory proves infinite homology 3-spheres in homology 4-spheres.
problem Existence of homology 3-spheres in homology 4-spheres.
method Diagrammatics of surface cross sections, Taubes' work.
result Infinite number of homology 3-spheres in homology 4-spheres.
The paper studies Finsler spheres with constant flag curvature and finite orbits of prime closed geodesics.
problem Investigating Finsler spheres with specific curvature properties and geodesic orbits.
method Analyzing the action of isometries and loops on Finsler spheres, focusing on finite orbits of prime closed geodesics.
result The existence of geometrically distinct orbits of prime closed geodesics and their properties.
In the paper we define a "volume" for simplicial complexes of flag tetrahedra. This generalizes and unifies the classical volume of hyperbolic manifolds and the volume of CR tetrahedra complexes. We describe when this volume belongs to the Bloch group. In doing so, we recover and generalize results of Neumann-Zagier, N…
A Finsler space is said to be geodesically reversible if each oriented geodesic can be reparametrized as a geodesic with the reverse orientation. A reversible Finsler space is geodesically reversible, but the converse need not be true. In this note, building on recent work of LeBrun and Mason, it is shown that a geodes…
We study a sigma-model with target space the flag manifold U(3)/U(1)^3. A peculiarity of the model is that the complex structure on the target space enters explicitly in the action. We describe the classical solutions of the model for the case when the worldsheet is a sphere CP^1.
Study improves Heegaard Floer homology relations for knots in homology spheres.
problem Improving relations in Heegaard Floer homology for knots in homology spheres.
method Proved inequality for d-invariants, used reduced Floer homology rank relations.
result Degree one maps between aspherical Seifert homology spheres are homotopic to homeomorphisms if Heegaard Floer homologies are isomorphic.
We show that among Seifert fibered integer homology spheres, Poincare sphere (with either orientation) is the only non-trivial example which has trivial Heegaard Floer homology. Together with an earlier result, this shows that if an integer homology sphere has trivial Heegaard Floer homology, then it is a connected sum…
Surgery on knots can produce non-separating spheres, using Heegaard Floer homology.
problem Conditions for surgery on knots to produce non-separating spheres.
method Heegaard Floer homology
result Sufficient conditions for a knot to be unknotted.
We present a systematic calculation of the volumes of compact manifolds which appear in physics: spheres, projective spaces, group manifolds and generalized flag manifolds. In each case we state what we believe is the most natural scale or normalization of the manifold, that is, the generalization of the unit radius co…
Surgery obstructions extended to integer homology spheres using Heegaard Floer homology.
problem Obstructing knots in integer homology spheres using surgery.
method Extending Heegaard Floer homology obstructions to all integer homology spheres for both positive and negative surgeries.
result Deduced a lower bound on b2(W) for smooth cobordism between integer homology spheres. The study connects polygon areas and projective structures in 3D space.
problem Relating polygon areas and projective structures in 3D space.
method Investigates positive tuples of complete flags in R^3 and their associated polygons in RP^2.
result Establishes a relationship between Holmes-Thompson area and projective structures.
Schottky groups constructed from flag manifolds' partial cyclic orders.
problem Constructing Schottky groups from geometric structures.
method Using 3-hyperconvexity and partial cyclic orders on flag manifolds, constructing Schottky groups. result Schottky groups correspond to positive representations in Fock and Goncharov's sense.
The paper identifies and illustrates families of knot diagrams yielding lens spaces from various homology spheres.
problem Identifying and illustrating families of knot diagrams yielding lens spaces from different homology spheres.
method Concrete knot diagrams and splicing of homology spheres.
result Families of lens space surgeries in various homology spheres are identified and illustrated.
We construct complexes P1n of Soergel bimodules which categorify the Young idempotents corresponding to one-column partitions. A beautiful recent conjecture of Gorsky-Rasmussen relates the Hochschild homology of categorified Young idempotents with the flag Hilbert scheme. We prove this conjecture for P1n an…