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.
Improved optimal regularity for harmonic almost complex structures.
problem Establishing optimal regularity for harmonic almost complex structures.
method Quantitative stratification method and rectifiability of singular strata.
result Optimal regularity theory for energy minimizing harmonic almost complex structures.
New methods compute partial dependence curves directly from data.
problem Limited accessibility and interpretability of partial dependence curves.
method Developed methods for numerical and categorical variables, directly from training data.
result Our methods provide a direct estimate of partial dependence without fitting models.
Let M be a compact smooth Riemannian n-manifold with boundary. We combine Gromov's amenable localization technique with the Poincaré duality to study the {\sf traversally generic} geodesic flows on SM, the space of the spherical tangent bundle. Such flows generate stratifications of SM, governed by rich univers…
We prove partial regularity of stationary solutions and minimizers u from a set Ω⊂Rn to a Riemannian manifold N, for the functional ∫ΩF(x,u,∣∇u∣2)dx. The integrand F is convex and satisfies some ellipticity and boundedness assumptions. We also develop a new monotonicity formula and…
We study singular hyperkahler quotients of the cotangent bundle of a complex semisimple Lie group as stratified spaces whose strata are hyperkahler. We focus on one particular case where the stratification satisfies the frontier condition and the partial order on the set of strata can be described explicitly by Lie the…
In many healthcare settings, intuitive decision rules for risk stratification can help effective hospital resource allocation. This paper introduces a novel variant of decision tree algorithms that produces a chain of decisions, not a general tree. Our algorithm, α-Carving Decision Chain (ACDC), sequentially carves o…
We consider Type I Ricci flows and obtain integral estimates for the curvature tensor valid up to, and including, the singular time. Our estimates partially extend to higher dimensions a curvature estimate recently shown to hold in dimension three by Kleiner and Lott. To do this we adapt the technique of quantitative s…
We fully describe the horofunction boundary ∂hL2 with the word metric associated with the generating set {t,at} (i.e the metric arising in the Diestel-Leader graph DL(2,2)). The visual boundary ∂∞L2 with this metric is a subset of ∂hL2. Although $\partial_\infty L_2…
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…
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.
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.
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.
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.
Alexandrov spaces have a special stratification that maps to spheres.
problem Characterizing the structure of Alexandrov spaces.
method Extremal stratification and space of directions analysis.
result Alexandrov spaces are homeomorphic to spheres in their space of directions.
Hidden stratification causes machine learning models to fail on rare but important patient subgroups.
problem Machine learning models fail on rare patient subgroups not identified during training or testing.
method Assessed techniques for measuring and describing hidden stratification effects on multiple medical imaging datasets.
result Evidence of hidden stratification leading to over 20% performance differences on clinically important subsets.
Investigates properties of moment maps and stratifications on Lie groups.
problem Understanding moment maps and stratifications on real reductive Lie groups.
method Functorial, algebraic approach to moment map and Kirwan-Ness stratification.
result Properties and properties of moment maps and stratifications established.
Optimizes biharmonic map regularity using stratification methods.
problem Improving the known almost optimal regularity of biharmonic maps.
method Quantitative stratification method.
result Optimal regularity results for minimizing biharmonic maps.
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 article consist of two main parts: an analog of the Leray Theory for Singular Varieties and its application to the Theory of Parshin's Residues. The first part is independent from the second. It uses the theory of Whitney stratifications. The second part is an application of the first. In particular, a geometric an…
The aim of this paper is to compare stratifications of moduli spaces given by group actions in the case of similarity of matrices introduced by Arnold and the author's stratification by projective orbifolds, and its relation to deformations o elements in the moduli space.
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.
Social media enhances or diminishes scientific status, depending on usage.
problem Impact of social media on scientific stratification and mobility.
method Logistic Attribution Analysis combining statistical and machine learning methods.
result Social media promotes stratification and mobility, but beyond a threshold, it negatively impacts status.
Study clarifies variance of stratification estimators for causal effects.
problem Estimating average causal effects with discrete covariates.
method Combines insights from potential outcomes, causal diagrams, and structural models.
result Derives expressions for the variance of stratification estimators.
We give a geometric proof of existence of Whitney stratifications of definable sets in o-minimal structures.
This paper provides a stratification of semi-algebraic sets in the plane with finitely many geodesic segments.
problem How to stratify semi-algebraic sets in the plane with finitely many geodesic segments.
method Develops a semi-algebraic stratification of a real semi-algebraic set in the plane with open cells having the finiteness property.
result Provides insights for high-dimensional stratifications of semi-algebraic sets in connection with geodesics.
We study the topology of the inertia space of a smooth G-manifold M where G is a compact Lie group. We construct an explicit Whitney stratification of the inertia space, demonstrating that the inertia space is a triangulable differentiable stratified space. In addition, we demonstrate a de Rham theorem for differ…
The paper stratifies projective measured laminations and identifies a group of transformations.
problem Stratifying the space of projective measured laminations.
method Introducing a natural stratification and proving rigidity results.
result The group of self-homeomorphisms preserving the stratification is identified with the extended mapping class group.
Study lifts Schubert stratification to Spinn+1, revealing new Bruhat cells.
problem Lifting Schubert stratification to Spinn+1. method Explicit parameterizations of Bruhat cells using minimal-length permutations.
result Stratification of Spinn+1 reveals new geometric structure. The paper describes a stratification of a compactified Hurwitz space using combinatorial trees.
problem Describing the boundary stratification of a compactified Hurwitz space.
method Using decorated trees to describe the boundary strata and their containment relations.
result The boundary strata of the compactified Hurwitz space are in bijection with decorated trees, and containment is given by edge contraction.
The paper studies harmonic map flows and proves rectifiability of singular sets.
problem Understanding the structure of singular sets in harmonic map flows.
method Investigates the stratification theory for suitable solutions using tangent measures.
result Each time slice of the singular set is rectifiable.
The complement of a complex hyperplane arrangement is known to be homotopic to a minimal CW complex. There are several approaches to the minimality. In this paper, we restrict our attention to real two dimensional cases, and introduce the "dual" objects so called minimal stratifications. The strata are explicitly descr…
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 …
The paper offers simple, near-optimal algorithms for multi-group learning.
problem Learning predictors within subgroups of a population, addressing fairness and hidden stratification.
method Studies the structure of solutions and provides simple, near-optimal algorithms.
result Simple and near-optimal algorithms for multi-group learning.
The paper studies degenerate and nonlocal elliptic operators on Poincaré-Einstein manifolds.
problem Understanding the singular sets of degenerate and nonlocal elliptic operators on Poincaré-Einstein manifolds.
method Developing quantitative differentiation theory, stratification, Minkowski estimates, and ε-regularity results.
result Uniform Hausdorff measure estimates for the singular sets of degenerate/singular elliptic operators.
The monster tower is a tower of spaces over a specified base; each space in the tower is a parameter space for curvilinear data up to a specified order. We describe and analyze a natural stratification of these spaces.
Decomposes smooth manifolds into algebraic submanifolds.
problem Understanding the structure of smooth manifolds induced by continuous selections.
method Generic continuous selection of smooth functions provides stratification of the manifold.
result Stratification leads to local topological structure with nondegenerate critical points.
Study chaotic dynamics in social stratification models leading to thermalization and turbulence.
problem Understanding social stratification dynamics through chaotic nonlinear systems.
method Modeling social network links with oscillators and energies, studying Hamiltonian evolution and nonlinear interactions.
result Chaotic dynamics leads to dynamical thermalization and Kolmogorov-Zakharov turbulence, with implications for wealth inequality.
Study almost rigidity of super Ricci flow with non-negative Muller quantity.
problem Almost rigidity properties of super Ricci flow with non-negative Muller quantity.
method Almost splitting and quantitative stratification theorems established by Bamler for Ricci flow.
result Obtained almost constancy for a certain integral quantity concerning scalar curvature at an almost self-similar point.
Study identifies and estimates treatment effect heterogeneity within principal stratification subpopulations.
problem Causal inference with intermediate outcomes and treatment effect heterogeneity.
method Proposes a novel doubly cross-fit doubly robust machine learner to efficiently learn conditional principal causal effects under principal ignorability.
result Demonstrates informative patterns of treatment effect heterogeneity within the always-survivor subpopulation in an acute lung injury trial.
We consider a Morse function f and a Morse-Smale gradient-like vector field X on a compact connected oriented 3-manifold M such that f has only one critical point of index 3. Based on Laudenbach's ideas, we will show that the flow of X can be isotoped into one so that the trajectory spaces of the new flow pro…
New method improves compatibility of risk stratification models without sacrificing accuracy.
problem Compatibility issues arise when updating clinical machine learning models.
method Proposes rank-based compatibility measure and new loss function.
result Increased compatibility of models by 0.019 with no loss in discriminative performance.
Study cohomology of abelian differentials, find new stratifications.
problem Cohomology classes in abelian differentials strata.
method Unified topological approach to find explicit stratifications.
result Explicit stratification of spin moduli space for odd spin structures.
A homology stratification is a filtered space with local homology groups constant on strata. Despite being used by Goresky and MacPherson [Intersection homology theory: II, Inventiones Mathematicae, 71 (1983) 77-129] in their proof of topological invariance of intersection homology, homology stratifications do not appe…
Survey on hyperplane arrangements and their topology.
problem Topology of hyperplane arrangements.
method Focus on the relationship between topology and real structure.
result Relationship between topology and real structure of hyperplane arrangements.
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.
We reduce variance in monetization metrics for ranking experiments.
problem Heavy-tailed monetization metrics lead to unreliable conclusions in A/B experiments.
method Post-stratification combined with CUPED.
result Significant reduction in variance and improved decision stability.
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.