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.
In the recent years, a number of issues concerning distributions generating 1- flags (called also Goursat flags) has been analyzed. Presently similar questions are discussed as regards distributions generating multi-flags. (In fact, only so-called special multi-flags, to avoid functional moduli.) In particular and fore…
This paper classifies strongly nilpotent special multi-flags and their Goursat counterparts.
problem Local classification of strongly nilpotent special multi-flags and Goursat distributions.
method Study of special multi-flags in homogeneous case, focusing on their weights.
result Strongly nilpotent germs of multiflags from different singularity classes are pairwise inequivalent.
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.
This paper constructs Brownian motion on complex flag manifolds and finds joint distribution of stochastic areas.
problem Modeling stochastic areas on complex partial flag manifolds.
method Constructs Brownian motion on complex partial flag manifolds and uses it to find joint distribution of stochastic areas.
result Limit law of stochastic areas is a multivariate Cauchy distribution.
In the paper we discuss certain classes of vector distributions in the tangent bundles to manifolds, obtained by series of applications of the so-called generalized Cartan prolongations (gCp). The classical Cartan prolongations deal with rank-2 distributions and are responsible for the appearance of the Goursat distrib…
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 paper studies invariant functions and their relation to Landsberg surfaces.
problem Investigating the geometry of invariant functions and their applications to Landsberg surfaces.
method Investigating the geometry of S-invariant functions and their associated vertical subdistribution, and relating the holonomy distribution to these subdistributions. result For Landsberg surfaces, if the flag curvature is S-invariant, it is constant, and the surface is Riemannian. We give definition of a holonomy flag in subRiemannian geometry --- a generalization of a Riemannian holonomy algebra --- and calculate it for the 3D subRiemannian Lie groups. We rewrite and give new interpretation for the Codazzi equations for the (2,3)-distributions on the SU(2) and the Heisenberg group.
Motivated by the geometric theory of differential equations and the variational approach to the equivalence problem for geometric structures on manifolds, we consider the problem of equivalence for distributions with fixed submanifolds of flags on each fiber. We call them flag structures. The construction of the canoni…
Random matrix ensembles yield uniform distributions on manifolds.
problem Understanding distributions of vectors in random matrix ensembles.
method Analyzing eigenvalues, singular values, and Autonne-Takagi vectors of various random matrix ensembles.
result Uniform distributions on specific manifolds for different types of random matrix ensembles.
EL framework certifies and flags bias in ML models without distributional assumptions.
problem Systematic performance disparities across sensitive subpopulations in ML models.
method Empirical likelihood-based approach for non-parametric fairness auditing.
result EL framework outperforms bootstrap methods in certification and subpopulation discovery.
Compactifies geodesic flows on hyperbolic surfaces, revealing attractive circles at infinity.
problem Geodesic flows on non-compact hyperbolic surfaces without cusps.
method Constructs a geometrical compactification using one-dimensional distributions tangent to stable and unstable horocycles.
result Existence of attractive circles at infinity in the compactified flow.
We solve the equivalence problem for rank 3 completely nonholonomic vector distributions with 6-dimensional square on a smooth manifold of arbitrary dimension n under very mild genericity conditions. The main idea is to consider the projectivization of the annihilator of a given 3-dimensional distribution. It is natura…
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.
A simple method flags images as out-of-distribution based on their distance to nearest neighbors.
problem Detecting images not aligned with a trained model's in-distribution data.
method Flag images as OOD if their average distance to K nearest neighbors is large in the classifier's representation space.
result Simple methods can outperform more complex ones when considering learned representations.
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.
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.
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.
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.
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.
Study equigeodesics on G2-type flag manifolds, splitting tangent spaces.
problem Characterize geodesics in G2-type flag manifolds. method Analyze flag manifolds with G2-type t-roots, split tangent spaces, and classify equigeodesics. result Characterized structural equigeodesic vectors in flag manifolds.
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 λ1-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.
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-transformations. result All invariant complex Dirac structures with constant real index on a maximal flag manifold are described.
We investigate the secant dimensions and the identifiablity of flag varieties parametrizing flag of sub vector spaces of a fixed vector space. We give numerical conditions ensuring that secant varieties of flag varieties have the expected dimension, and that a general point on these secant varieties is identifiable.
The aim of this paper is to show that any stable complete Riemannian flag on a compact and connected manifold is conjugated to a flag of homogenus foliations (see Definitions). Also, we give a characterization of Riemannian flags that homogenus. This result is a step toward the classification of Riemannian flags.
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. Study spin chains and sigma models on flag manifolds, calculating spectra and geodesics.
problem Understanding the spectrum and geodesics of sigma models on flag manifolds.
method Connecting SU(n) spin chains to sigma models and calculating spectra and geodesics.
result Calculated the spectrum of the Laplace-Beltrami operator and geodesics for CP1 and F3. SLUG method detects bias and out-of-distribution content in generative models.
problem Generative models can underrepresent certain groups and fail on out-of-distribution data.
method SLUG: A new uncertainty quantification method for VAEs combining Laplace approximations and stochastic trace estimators.
result SLUG's UQ score correlates with bias and out-of-distribution content.
The paper explores curvature positivity on Kähler and quasi-Kähler flag manifolds.
problem Analyzing curvature positivity on specific geometric structures.
method Investigation of Griffiths and dual-Nakano positivity for curvature of Chern connections on Kähler and quasi-Kähler flag manifolds.
result Classification of Kähler flag manifolds with Griffiths semi-positive curvature and restrictions for quasi-Kähler flag manifolds.
In the first part of this paper we study geometric formality for generalized flag manifolds, including full flag manifolds of exceptional Lie groups. In the second part we deal with the problem of the classification of invariant almost complex structures on generalized flag manifolds using topological methods.
The paper explores conditions for Finsler surfaces to be Landsbergian and classify surfaces with specific flag curvature conditions.
problem Conditions for Finsler surfaces to be Landsbergian and classify surfaces with specific flag curvature conditions.
method Investigates geometric objects associated with the global Berwald distribution and studies Finsler surfaces satisfying certain flag curvature conditions.
result Classifies Finsler surfaces with specific flag curvature conditions and shows they are Riemannian under certain conditions.
In this work we prove a Baum-Bott type residue theorem for flags of holomorphic foliations. We prove some relations between the residues of the flag and the residues of their correspondent foliations. We define the Nash residue for flags and we give a partial answer to the Baum-Bott type rationality conjecture in this …
We study an intrinsic distribution, called polar, on the space of l-dimensional integral elements of the higher order contact structure on jet spaces. The main result establishes that this exterior differential system is the prolongation of a natural system of PDEs, named pasting conditions, on sections of the bundle…
A flag area measure on an n-dimensional euclidean vector space is a continuous translation-invariant valuation with values in the space of signed measures on the flag manifold consisting of a unit vector v and a (p+1)-dimensional linear subspace containing v with 0≤p≤n−1. Using local parallel sets, …
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.
The paper describes invariant twisted Kähler-Einstein metrics on flag varieties.
problem Existence and properties of invariant twisted Kähler-Einstein metrics on flag varieties.
method Invariant twisted Kähler-Einstein metrics on flag varieties, exploring applications and inequalities.
result Established inequalities related to optimal volume upper bounds for Kähler metrics.
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 flag is a sequence of nested subspaces. Flags are ubiquitous in numerical analysis, arising in finite elements, multigrid, spectral, and pseudospectral methods for numerical PDE; they arise in the form of Krylov subspaces in matrix computations, and as multiresolution analysis in wavelets constructions. They are comm…
If the flag curvature of a Finsler manifold reduces to sectional curvature, then locally either the Finsler metric is Riemannian, or the flag curvature is isotropic.
One of the most important problems in Finsler geometry is to classify Finsler metrics of scalar flag curvature. In this paper, we study the classification problem of Randers metrics of scalar flag curvature. Under the condition that β is a Killing 1-form, we obtain some important necessary conditions for Randers metr…
Note refutes examples of Landsberg surfaces with vanishing flag curvature.
problem Verifying examples of Landsberg surfaces with specific curvature properties.
method Analyzing examples from Zhou's result to show they are Berwaldian.
result Examples of Landsberg surfaces with vanishing flag curvature are Berwaldian.
Study flag curvature in homogeneous Finsler spaces with a specific metric.
problem Analyzing flag curvature in homogeneous Finsler spaces with a generalized m-Kropina metric. method Provided explicit formula for flag curvature, showed equivalence of definitions, and studied curvature of naturally reductive spaces.
result Equivalence of two definitions of naturally reductive homogeneous Finsler spaces for the generalized m-Kropina metric.