New stratification reveals intrinsic singularity types of orbit spaces.
problem Understanding the intrinsic structure of orbit spaces under Lie group actions.
method Introduced the isostabilizer decomposition and established a map to Klein strata.
result A new canonical stratification on the manifold clarifies the relationship with classical structures.
The moduli space of Higgs bundles is stratified into complex symplectic submanifolds.
problem Constructing a complex Whitney stratification of the moduli space of Higgs bundles.
method Showed that the orbit type decomposition is a complex Whitney stratification with each stratum being a complex symplectic submanifold.
result The moduli space of Higgs bundles is a stratified complex symplectic space.
We show that, if the family \cal{O} of orbits of all vector fields on a subcartesian space P is locally finite and each orbit in \cal{O} is locally closed, then \cal{O} defines a smooth Whitney A stratification of P. We also show that the stratification by orbit type of the space M/G of orbits of a proper action of a L…
Let G be a Lie group, and let (M,ω) be a symplectic manifold. If G admits a Hamiltonian action on (M,ω) with momentum map μ, then M, the zero-level set of μ, the orbit space, and the corresponding symplectic quotient all have induced stratifications. We push this setting into the language of differential …
Maps vector fields between stacks and orbit spaces.
problem Understanding vector fields on stacks and orbit spaces.
method Morita stratifications and geometric vector fields correspondence.
result Derives stacky version of Gauss lemma and extends Palais' theorem.
New criteria found for Willmore submanifolds in Lie group orbits.
problem Criteria for Willmore submanifolds in Lie group orbits.
method Criteria for Willmore submanifolds based on orbit type stratification.
result Found Willmore orbits in each stratified subset of orbit type.
Clarifies the structure of quantum states using algebraic methods.
problem Unclear stratification of quantum states in physics literature.
method Analyzes the state space S(A) of a finite-dimensional C*-algebra A, focusing on unitary orbits and their properties.
result Identifies a natural Whitney stratification of the state space into matrices of fixed rank, providing a pseudo-manifold structure.
For a principal $\rmSU(n)$-bundle over a compact manifold of dimension 2,3,4, we determine the orbit types of the action of the gauge group on the space of connections modulo pointed local gauge transformations. We find that they are given by Howe subgroups of $\rmSU(n)$ for which a certain characteristic equation is…
We introduce the notions of a differentiable groupoid and a differentiable stratified groupoid, generalizations of Lie groupoids in which the spaces of objects and arrows have the structures of differentiable spaces, respectively differentiable stratified spaces, compatible with the groupoid structure. After studying b…
Geometric invariant theory for real Lie groups proved.
problem Closed orbits and null cone stratification in real reductive Lie groups.
method Completely self-contained proof focusing on geometric and analytic methods.
result Applies to non-rational linear actions.
Slice theorem in infinite dimensions for Lie groups.
problem Generalizing slice theorem to infinite-dimensional settings.
method Developed slice theorem for locally convex Lie groups on locally convex manifolds using advanced theorems.
result Existence of orbit type stratification under slice condition.
New invariant csm simplifies computing geometric invariants of recursive group orbits.
problem Computing geometric invariants of recursive group orbits is hard.
method Introduced new invariant csm and used it to compute invariants explicitly. result Explicit formulas for local Euler obstructions and sectional Euler characteristics.
We will provide a lower bound for the equivariant Lusternik-Schnirelmann category of an arbitrary proper action in terms of the stratification by orbit types, and an upper bound for proper polar actions in terms of the equivariant Lusternik-Schnirelmann category of its generalized Weyl group. As an application we repro…
Study smooth structures and stratifications on orbit spaces of Lie groupoids.
problem Understanding the smooth structure and stratification of orbit spaces of Lie groupoids.
method Utilizing Spallek's framework and extending classical theories of proper Lie group actions.
result Established Morita invariance of smooth structures and stratifications on orbispaces.
Diffeological and differential spaces are generalisations of smooth structures on manifolds. We show that the "intersection" of these two categories is isomorphic to Frölicher spaces, another generalisation of smooth structures. We then give examples of such spaces, as well as examples of diffeological and differential…
Study of tangent spaces in diffeological spaces under Lie group actions.
problem Understanding tangent spaces in generalized spaces.
method Generalized tangent space construction and isomorphism proof.
result Internal tangent space isomorphic to stratified tangent space.
Study shows convex contact spheres resemble contact ellipsoids.
problem Characterizing the structure of convex contact spheres.
method Stratification by Reeb orbit periods and analysis of spectral invariants.
result Any stratum of a convex contact sphere is an integral homology sphere, and spectral invariants coincide with action values.
Let Σ be a closed surface, G a compact Lie group, not necessarily connected, with Lie algebra g, endowed with an adjoint action invariant scalar product, let ξ:P→Σ be a principal G-bundle, and pick a Riemannian metric and orientation on Σ so that the corresponding Yang-Mills equations are defined.…
Study solutions to metrizability equation on manifolds.
problem Finding solutions to a differential equation for pseudo-Riemannian metrics.
method Analyzing generic conditions on the prolonged system of the metrizability equation.
result Solutions stratify manifolds based on strict signature of metrics.
In this paper, we study geometric properties of quotient spaces of proper Lie groupoids. First, we construct a natural stratification on such spaces using an extension of the slice theorem for proper Lie groupoids of Weinstein and Zung. Next, we show the existence of an appropriate metric on the groupoid which gives th…
Generalizes Landau-Ginzburg mirrors for Frobenius manifolds in Dynkin type A.
problem Classifying Frobenius manifold structures in Dynkin type A.
method Generalizing the method from previous works, developing a pole-collision framework.
result Structural result at the level of prepotential for arbitrary rank and dimension.
We revisit Atiyah and Bott's study of Morse theory for the Yang-Mills functional over a Riemann surface, and establish new formulas for the minimum codimension of a (non-semi-stable) stratum. These results yield the exact connectivity of the natural map (C_{min} E)//G(E) --> Map^E (M, BU(n)) from the homotopy orbits of…
This paper studies geodesics between covariance matrices of different ranks using the Bures-Wasserstein metric.
problem Geodesics between covariance matrices of varying ranks.
method Analyzes the Bures-Wasserstein distance on covariance matrices, completing previous work on geodesics and providing explicit formulas.
result The set of all minimizing geodesics between two covariance matrices is parametrized by a closed unit ball in R(k−r)imes(l−r). Combines k-means and hill climbing for stratification and allocation.
problem Optimizing stratification and sample allocation for complex surveys.
method Combining k-means type algorithms with hill climbing.
result Multi-stage combination algorithms generally perform well compared to recent methods.
Integral estimates for Ricci flows up to singular time.
problem Integral estimates for curvature tensor in Type I Ricci flows.
method Adapted quantitative stratification technique.
result Partial extension of curvature estimates to higher dimensions.
We prove analogues for Cartan geometries of Gromov's major theorems on automorphisms of rigid geometric structures. The starting point is a Frobenius theorem, which says that infinitesimal automorphisms of sufficiently high order integrate to local automorphisms. Consequences include a stratification theorem describing…
The study classifies rational 1-forms on the Riemann sphere with simple poles.
problem Classifying rational 1-forms on the Riemann sphere with specified pole conditions.
method Recognized three equivalent atlases, proved submanifold properties, and used PSL(2,C) action.
result Quotients of isochronous 1-forms admit stratified orbit types.
We develop the Lorentzian geometry of a crooked halfspace in 2+1-dimensional Minkowski space. We calculate the affine, conformal and isometric automorphism groups of a crooked halfspace, and discuss its stratification into orbit types, giving an explicit slice for the action of the automorphism group. The set of parall…
Paper introduces a new geometric homology theory and applies it to Gromov-Witten theory.
problem Developing a new homology theory for orbifolds with corners.
method Using stratification and triangulation theories of Lie groupoids and their orbit spaces, extending to Lie groupoids with corners.
result Proposes and proves the geometric homology theory (GHT), a flexible generalization of singular homology.
We list up all the possible local orbit types of hyperbolic or elliptic orbits for the isotropy representations of semisimple pseudo-Riemannian symmetric spaces. It is key to give a recipe to determine the local orbit types of hyperbolic principal orbits by using three kind of restricted root systems and Satake diagram…
Deep learning model creates patient representations for scalable EHR-based stratification.
problem Challenges in summarizing and representing patient data from EHRs prevent scalable stratification analysis.
method Unsupervised framework based on deep learning (ConvAE) using word embeddings, CNNs, and autoencoders.
result ConvAE significantly outperformed baselines in clustering diverse patient cohorts, identifying clinically relevant subtypes.
Develops Poisson and Dirac manifolds of compact types with applications.
problem Understanding Poisson and Dirac manifolds of compact types.
method Establishing structural results, local normal forms, canonical stratifications, and Weyl type resolutions.
result Every Poisson manifold of compact type is necessarily regular.
We prove the existence of Verdier stratifications for sets definable in any o-minimal structure on (R, +, .). It is also shown that the Verdier condition (w) implies the Whitney condition (b) in o-minimal structures on (R, +, .). As a consequence the Whitney Stratification Theorem holds. The existence of (wf)-stratific…
Study geodesic orbit property on pseudo-Riemannian H-type nilmanifolds.
problem Characterize geodesic orbit property for pseudo-Riemannian H-type Lie groups.
method Extend results from Riemannian to pseudo-Riemannian H-type Lie groups, focusing on minimal admissible Clifford modules.
result Complete characterization of geodesic orbit property for pseudo-Riemannian H-type Lie groups.
The paper studies HKKN stratifications for non-compact spaces and proves convexity properties.
problem Proving convexity properties of moment maps for non-compact subsets.
method Algebraic and analytical study of HKKN stratifications for a vector space and compact Kähler manifold, then applying to non-compact subsets.
result Convexity properties of moment maps for invariant subsets are proven.
Study closed G2-structures with T3-symmetry, classifying them into types and deriving hypersymplectic structures.
problem Classify closed G2-structures with T3-symmetry and derive associated hypersymplectic structures.
method Decompose G2-structures into canonical forms, classify structures based on orbit isotropy, and derive hypersymplectic structures.
result Closed G2-structures with T3-symmetry are classified into two types, leading to specific hypersymplectic structures.
{\bf Construction.} For a dominating polynomial mapping {F:Kn→Kl} with an isolated critical value at 0 (K an algebraically closed field of characteristic zero) we construct a closed {\it bundle} GF⊂T∗Kn. We restrict GF over the critical points Sing(F) of F in F−1(0) and partiti…
This paper stratifies spaces of locally convex curves by their itineraries.
problem Homotopy type of spaces of locally convex curves with prescribed initial and final jets.
method Defining itineraries and proving submanifolds of finite codimension.
result Stratification of the space of curves by their itineraries.
The natural partial ordering of the orbit types of the action of the group of local gauge transformations on the space of connections in space-time dimension d<=4 is investigated. For that purpose, a description of orbit types in terms of cohomology elements of space-time, derived earlier, is used. It is shown that, on…
The study confirms a conjecture about polynomials related to symmetric spaces.
problem Understanding polynomials associated with isotropy orbits of symmetric spaces.
method Identified Reiswich's polynomials as special cases of Jacobi polynomials and proved their conjecture.
result The polynomials have pairwise different real roots in the interval [0,1].
This paper provides a ML framework for diabetes prediction and care management.
problem Diabetes prediction and care management challenges in real-world healthcare.
method Illustrates a Machine Learning framework for T2DM prediction and risk stratification.
result ML models align with physician's disease management steps.
Stratifies representation varieties of twisted Hopf links.
problem Stratifying representation varieties of twisted Hopf links.
method Using stratification of AGLr(C)-representation varieties of the fundamental group of the complement of a twisted Hopf link. result Explicit description and computation of motives for ranks 1 and 2.
Study finds homogeneous spaces with geodesic orbits but no integrable distributions.
problem Characterizing homogeneous spaces with specific geometric properties.
method Examined Lie groups with compact stabilizers and classified spaces based on geodesic orbits and integrable distributions.
result Identified homogeneous spaces with geodesic orbits but lacking integrable invariant distributions.
The paper defines a stratification for Lie groupoids in a tame topology context.
problem Presenting a tame topology counterpart to canonical stratification of Lie groupoids.
method Using Shiota's isotopy lemma and approximation theorem, the paper defines a canonical Whitney stratification of definable Lie groupoids into invariant strata.
result A canonical Whitney stratification of the Lie groupoid into definable strata invariant under the groupoid action.
Generalizes results for lambda-connections and Higgs bundles.
problem Understanding the Bialynicki-Birula stratification of lambda-connections.
method Analyzes the Bialynicki-Birula decomposition and its relation to Morse and partial oper stratifications.
result Fibers of the Morse and partial oper stratifications are transverse at the base point and are half-dimensional affine spaces.
A Lie group G naturally acts on its Lie algebra ≫, called the adjoint action. In this paper, we determine the orbit types of the compact exceptional Lie group G2 in its Lie algebra ≫2. As results, the group G2 has four orbit types in the Lie algebra ≫2 as $$ G_2/G_2, \quad G_2/(U(1) \times U(1)), …
Paper confirms MCS spaces are equivalent to CS sets.
problem Understanding the equivalence of MCS and CS sets.
method Analyzing the MCS stratification and intrinsic stratification.
result MCS spaces are equivalent to CS sets with respect to their stratification.
Geometrically connects adjoint and coadjoint orbits of semidirect products.
problem Establishing a bijection between adjoint and coadjoint orbits for semidirect products.
method Proving a geometric bijection using specific subgroup conditions and homotopy types.
result Homotopy types of orbits in bijection are the same under certain conditions.