Generalizes soft noncommutative schemes to flag varieties.
problem Applying soft noncommutative schemes to flag varieties.
method Generalization via toric geometry and distinguished affine charts.
result Soft noncommutative schemes can be applied to flag varieties.
The study examines closed manifolds with ray nil-affine structures and their completeness.
problem Characterizing closed manifolds with ray nil-affine structures.
method Analyzing properties of nilpotent spaces and flag manifolds, applying rank conditions and automorphism group actions.
result Closed manifolds are either complete or their developing map is a cover onto the complement of a nil-affine subspace.
We show that a large class of non-metric, non-symplectic affine holonomies can be realized, uniformly and without case by case considerations, by Weyl connections associated to the natural AHS-structures on certain generalized flag manifolds.
The paper explores domains in flag manifolds with specific geometric properties.
problem Exploring bounded domains in flag manifolds with co-compactly acting automorphism groups.
method Analyzing projective automorphism groups and convexity conditions.
result No examples of bounded domains exist in many flag manifolds, but they do in some cases.
For spherical Tits buildings of the classical types there are well-known explicit descriptions as flag complexes. Similarly for affine buildings of the classical types there are explicit constructions in terms of lattices. In this article we generalize the flag complex description to twin cities, a generalization of tw…
Study on Funk geometry volume growth and polytope flags, verifying conjectures.
problem Volume minimization in Funk geometry and polytopes.
method Analyzing Holmes--Thompson volume, studying asymptotics, computing coefficients.
result Second highest volume coefficient minimized by unique center point, maximized by regular polygons.
Researchers provide coordinates for Lie group configurations, linking to Teichmüller space and knot representations.
problem Finding coordinates for Lie group configurations.
method Explicit Fock-Goncharov coordinates for triples and quadruples of affine flags in rank 2 Lie groups.
result Explicit coordinates on higher Teichmüller space and boundary-unipotent representations of 3-manifold groups.
We develop an algebraic version of Cartan method of equivalence or an analog of Tanaka prolongation for the (extrinsic) geometry of curves of flags of a vector space W W W with respect to the action of a subgroup G G G of the G L ( W ) GL(W) G L ( W ) . Under some natural assumptions on the subgroup G G G and on the flags, one can pass from th…
Categorifies Hecke algebra subalgebra using sheaves and bimodules.
problem Categorify maximal commutative subalgebra of type A Hecke algebra.
method Construct monoidal functor from coherent sheaves on flag Hilbert scheme to Soergel bimodules, match Hochschild homology with sheaf Euler characteristics.
result Categorified projectors correspond to dg algebras of functions on affine charts of flag Hilbert schemes, conjecturally matching homology of g l N \mathfrak{gl}_N gl N projectors. The paper introduces a new spray deformation and finds new metrics.
problem Characterizing metrizability of new spray structures.
method Introducing one form deformation of sprays and analyzing their metrizability.
result Obtained new projectively flat metrics of constant flag curvature 1.
We consider faithful projective actions of a cocompact lattice of SL(2,R) on the projective plane, with the following property: there is a common fixed point, which is a saddle fixed point for every element of infinite order of the the group. Typical examples of such an action are linear actions, ie, when the action ar…
Study of parabolic Higgs bundles on curves with special fixed points.
problem Understanding fixed points of C i m e s \mathbb{C}^ imes C i m es -action on moduli spaces of Higgs bundles. method Analyzing C i m e s \mathbb{C}^ imes C i m es -action on moduli spaces, classifying fixed points, and studying Bialynicki-Birula flows. result Classification of very stable fixed points and their relation to Hitchin maps.
The study characterizes Finsler metrics and proves their rigidity.
problem Characterizing and proving rigidity of Busemann convex Finsler metrics.
method Proving Finsler metrics are nonpositively curved if and only if they are affinely equivalent to a Riemannian metric of nonpositive sectional curvature.
result Finsler metrics are precisely Berwald metrics of nonpositive flag curvature.
Research characterizes intersection cohomology groups of gauge theories and cotangent bundles.
problem Characterizing intersection cohomology groups of Coulomb branch gauge theories.
method Uses geometric Satake correspondence for Kac-Moody settings.
result Sketches proof of conjecture in affine type A.
Let G G G be a linear connected complex reductive Lie group. The purpose of this paper is to give explicit symplectic isomorphisms from twisted cotangent bundles of the complex generalized flag varieties, whose transition functions are given by affine transformations instead of linear transformations, onto the complex co…
The trace of the affine Hecke category is compared with the elliptic Hall algebra.
problem Comparing the trace of the affine Hecke category with the elliptic Hall algebra.
method Using Wakimoto objects and Rouquier complexes, the trace is generated by objects E e x t b f d E_{ extbf{d}} E e x t b f d . result The trace of the affine Hecke category yields an integral form A ~ \widetilde{\mathcal{A}} A of the elliptic Hall algebra. Let W ⋉ L W\ltimes L W ⋉ L be an irreducible affine Weyl group with Coxeter complex Σ Σ Σ , where W W W denotes the associated finite Weyl group and L L L the translation subgroup. The Steinberg torus is the Boolean cell complex obtained by taking the quotient of Σ Σ Σ by the lattice L L L . We show that the ordinary and flag h h h -polynomial…
Defines and studies positive configurations in affine buildings.
problem Understanding the geometry of positive configurations in affine buildings.
method Elementary definition and study of properties, conjectures involving minimal networks and max-flow/min-cut theorem.
result Conjectures about the tropicalized canonical functions in terms of the geometry of affine buildings.
This article is an exposition of four loosely related remarks on the geometry of Finsler manifolds with constant positive flag curvature. <p> The first remark is that there is a canonical Kahler structure on the space of geodesics of such a manifold. <p> The second remark is that there is a natural way to construct a (…
The paper studies Coulomb branches of quiver gauge theories and their connections to Grassmannians.
problem Understanding the structure of Coulomb branches in quiver gauge theories.
method Analyzes the moduli space of rational maps and slices in the affine Grassmannian.
result Identifies the quantized Coulomb branch with the truncated shifted Yangian.
The paper generalizes PCA to manifolds using barycentric subspaces.
problem Generalizing PCA to non-Euclidean spaces.
method Introducing barycentric subspaces and optimizing AUV criterion.
result Barycentric Subspaces Analysis (BSA) generalizes PCA to Riemannian manifolds.
Geometric methods for surface group representations in higher rank SL(2m+1,R).
problem Representations of surface groups in higher rank SL(2m+1,R).
method Para-complex and pseudo-Riemannian geometric techniques.
result One-to-one correspondence between Higgs bundles and isotropic P-alternating surfaces.
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.
Cominuscule subvarieties found in flag varieties.
problem Identifying special subvarieties in flag varieties.
method Using Dynkin diagrams to compute subvariety structure.
result Every flag variety has a cominuscule subvariety.
Investigates secant dimensions and identifiability in flag varieties.
problem Secant dimensions and identifiability of flag varieties.
method Numerical conditions ensuring secant varieties have expected dimension and points are identifiable.
result Secant varieties of flag varieties have expected dimension and points are identifiable under certain conditions.
Study of weighted nonlinear flags in symplectic geometry.
problem Understanding the geometry of weighted nonlinear flags.
method Generalizing weighted nonlinear Grassmannians to Frechet manifolds and using them to describe coadjoint orbits.
result Description of coadjoint orbits of Hamiltonian diffeomorphisms using weighted isotropic nonlinear flags.
New algorithm computes flag mean and median on flag manifolds.
problem Computing first order flag statistics on flag manifolds.
method Transformed problem to Stiefel manifold for optimization.
result Proved convergence and effectiveness of the flag-mean computation.
The paper studies Finsler manifolds with a new curvature concept.
problem Understanding Finsler manifolds with positive weighted flag curvature.
method Introducing a new curvature concept based on the flag curvature and a non-Riemannian quantity, T-curvature.
result Positive weighted flag curvature implies the manifold is diffeomorphic to Euclidean space.
The paper explores geometric formality and complex structures on flag manifolds.
problem Classifying invariant almost complex structures on flag manifolds.
method Geometric formality study and topological methods.
result Classification of invariant almost complex structures on flag manifolds.
Residues theorem for flags of holomorphic foliations proved.
problem Proving residue theorems for flags of holomorphic foliations.
method Defining Nash residue and proving relations between residues of flags and foliations.
result Partial answer to the rationality conjecture in this context.
Study nonlinear flags as coadjoint orbits of Hamiltonian diffeomorphisms.
problem Geometry of nonlinear flags and their coadjoint orbits.
method Generalization of nonlinear Grassmannians to Frechet manifolds.
result Description of symplectic nonlinear flags as coadjoint orbits.
The paper finds new invariant Einstein metrics on specific flag manifolds.
problem Finding Einstein metrics on generalized flag manifolds.
method Explicit construction and algebraic reduction of the Einstein equation.
result New invariant Einstein metrics on S p ( n ) Sp(n) S p ( n ) and S O ( 2 n ) SO(2n) S O ( 2 n ) flag manifolds. The paper classifies Randers metrics with scalar flag curvature.
problem Classifying Finsler metrics of scalar flag curvature.
method Investigating Randers metrics under the condition that β is a Killing 1-form.
result Obtained necessary conditions for Randers metrics to be of scalar flag curvature.
Minimal dimensions found for flag manifolds embeddings.
problem Finding the smallest dimensions for flag manifolds embeddings.
method Equivariant embeddings of orthogonal and unitary groups acting on real and complex flag manifolds.
result Minimal dimensions achieved at isospectral models.
Study equigeodesics on G 2 G_2 G 2 -type flag manifolds, splitting tangent spaces.
problem Characterize geodesics in G 2 G_2 G 2 -type flag manifolds. method Analyze flag manifolds with G 2 G_2 G 2 -type t t t -roots, split tangent spaces, and classify equigeodesics. result Characterized structural equigeodesic vectors in flag manifolds.
Extends residue theory to flags of holomorphic distributions.
problem Calculating the residue class of flags of holomorphic distributions.
method Developed an effective method to calculate the class in certain cases.
result Established a relation between degrees, tangency order, Euler characteristic, and curve degree.
The paper introduces geometric surgeries for flag structures and provides examples of uniformizable and non-uniformizable types.
problem Understanding and classifying flag structures of different types.
method Introducing geometric surgeries for flag structures and applying them to examples.
result Examples of both uniformizable and non-uniformizable flag structures were provided.
Researchers prove formulas for flag area measures, extending previous work.
problem Proving additive kinematic formulas for flag area measures.
method Introducing an algebraic framework to compute these formulas explicitly.
result Existence and explicit computation of additive kinematic formulas for flag area measures.
New recursive relations for symmetries of special 2-flags discovered.
problem Local finite classification and continuous numerical modulus of special 2-flags.
method Recursive relations derived for all infinitesimal symmetries of special 2-flags.
result Existence of a continuous numerical modulus of special 2-flags in length seven.
Study finds numerical moduli in special 2-flags of length 5.
problem Identifying numerical moduli in local classifications of special multi-flags.
method Analyzing distributions generating special multi-flags, focusing on lengths up to 5.
result Three numerical moduli appear in special 2-flags of length 5.
Study finds conditions for Kähler-Einstein metrics on flag manifolds.
problem Characterizing Kähler-Einstein metrics on flag manifolds.
method Using Lie theoretic data, establish a sufficient and necessary condition for λ 1 λ_1 λ 1 -extremality. result Identifies criteria for a metric to be a critical point of the first eigenvalue functional.
The paper calculates expected distances on partially oriented flag manifolds.
problem Understanding distances on partially oriented flag manifolds.
method Computing expected distances on low-dimensional examples.
result Computed expected distances on partially oriented flag manifolds.
The paper examines the geometry of specific submanifolds in flag manifolds.
problem Understanding the geometry of invariant almost semi Kähler submanifolds.
method Analyzing homogeneous spaces as almost Hermitian submanifolds of flag manifolds.
result Minimal and totally geodesic properties of certain submanifolds.
Conditions found for flag manifolds to use Bochner coordinates.
problem Finding conditions for Bochner coordinates on flag manifolds.
method Analyzing complex coordinates on flag manifolds of classical groups.
result Necessary and sufficient conditions for Bochner coordinates on flag manifolds.
Optimizes over flag manifolds for numerical PDE and statistics.
problem Optimizing over flag manifolds for numerical PDE and statistics.
method Develops tools for Riemannian optimization on flag manifolds, deriving analytic expressions and parameterizations.
result Closed-form analytic expressions and parameterizations for various geometric objects on flag manifolds.
New flag area measures constructed via integration over differential forms.
problem Constructing new flag area measures in high-dimensional spaces.
method Using local parallel sets and integration over the normal cycle of differential forms.
result Flag area measures span the space of all smooth SO(n)-covariant flag area measures.
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.
The paper classifies complex Dirac structures on flag manifolds.
problem Classifying invariant complex Dirac structures on flag manifolds.
method Described using roots of the Lie algebra and classified under B B B -transformations. result All invariant complex Dirac structures with constant real index on a maximal flag manifold are described.