New concept of proper-calibeating extends classic calibrated forecasts to proper scoring rules.
problem Defining and extending calibrated forecasts to proper scoring rules.
method Extending the concepts of calibrated and calibeating forecasts to proper scoring rules and proving their properties.
result Proper-calibration always implies calibration, but proper-calibeating does not necessarily imply calibeating.
New method for simplifying knots with specific properties.
problem Understanding knots with a specific unknotting number.
method Derive and apply the Montesinos trick for proper rational tangle replacement.
result Prove that knots with proper rational unknotting number one are prime and classify certain types.
Proper Lie groupoids have real analytic structures.
problem Finding analytic structures for proper Lie groupoids.
method Demonstrating the existence of a compatible real analytic structure.
result Proper Lie groupoids have real analytic structures.
Paper extends Calabi's extremal metric existence to compact Kähler manifolds.
problem Existence of Calabi's extremal metric on compact Kähler manifolds.
method Adapting recent breakthroughs on constant scalar Kähler metrics to extremal case, proving properness of modified Mabuchi energy.
result Existence of extremal metric with extremal vector V if and only if modified Mabuchi energy is proper. The paper explores proper actions and their relation to representation theory, with new quantitative methods.
problem Understanding proper actions and their connection to representation theory.
method Geometric criteria, sharpness measure, and dynamical volume estimates.
result New quantitative methods have established temperedness criteria for unitary representations.
Classifies π1-injective maps between non-compact surfaces.
problem Characterizing maps with injective fundamental groups.
method Proper homotopy classification of maps.
result All π1-injective proper maps are classified. Study Mabuchi metrics on Fano manifolds proving their existence and properness.
problem Existence and properness of Mabuchi metrics on Fano manifolds.
method Prove existence using properness of modified Ding functional and inverse implication.
result Establish criterion for Mabuchi metrics existence on Fano group compactifications.
We show that immersed minimal surfaces of R3 with bounded curvature and proper self intersections are proper. We also show that the restriction of the immersing map to a wide component is always proper. When the immersing map is injective the whole surface is a wide component. Prior to these results it wa…
Paper develops proper, lower-bounded losses for weakly supervised classification.
problem Weakly supervised classification with corrupted labels.
method Representation theorem for proper losses, derived condition for lower-boundedness, generalized logit squeezing.
result Proper and lower-bounded losses for weak-label learning.
Non-proper surface group action on product of trees found.
problem Proper surface group action on product of trees proposed, but not proper.
method Demonstrated non-properness of the surface group action.
result Surface group action on product of trees is not proper.
Analytic linearization and holomorphic extensions for proper groupoids.
problem Analytic linearization and holomorphic extensions of proper groupoids.
method Establish analytic linearization around invariant submanifolds and apply to holomorphic extensions.
result Proper groupoids admit holomorphic extensions.
Classifies non-linear Fredholm maps linking to stable homotopy groups of spheres.
problem Classifying non-linear proper Fredholm maps between Hilbert spaces.
method Using stable homotopy groups of spheres to classify maps up to proper homotopy.
result Determines the non-trivial kernel of the map from stable homotopy groups to non-linear proper Fredholm maps.
A novel framework quantifies uncertainty using proper scores for various tasks.
problem Uncertainty quantification in machine learning for reliable applications.
method Proposes a general framework based on proper scores for epistemic, aleatoric uncertainty, and model calibration.
result Achieves state-of-the-art uncertainty estimation for large language models and generative models.
First example of open manifold with positive Ricci curvature and non-proper Busemann function.
problem Counterexample to Busemann function properness in open manifolds with nonnegative Ricci curvature.
method Provided an open manifold with positive Ricci curvature and non-proper Busemann function.
result First example of open manifold with positive Ricci curvature and non-proper Busemann function.
A study of proper affine vector fields in plane symmetric static space-times by using the rank of the Rieman matrix and holonomy. Studying proper affine vector fields in each case, It is shown that the special class of the above space-times admit proper affine vector fields.
Paper analyzes proper losses and their performance in machine learning tasks.
problem Understanding the performance of estimators and forecasters in machine learning tasks.
method Analyzes surrogate regret and convergence rates for strictly proper losses.
result Strongly proper losses achieve the optimal convergence rate.
First proper learning algorithm for Gaussian halfspaces with matching sample and computational complexity.
problem Agnostically learning halfspaces under Gaussian distribution.
method First proper learning algorithm with matching sample and computational complexity.
result First proper learning algorithm for agnostically learning halfspaces under Gaussian distribution with matching sample and computational complexity.
Proper Lie groupoids can be desingularized to regular ones.
problem Desingularizing proper Lie groupoids to regular ones.
method Successive blow-up construction on a proper Lie groupoid.
result Desingularization of proper Lie groupoids to regular ones, arbitrarily close in Gromov-Hausdorff distance.
Authors create stable proper biharmonic maps from unit ball to spheres.
problem Constructing stable proper biharmonic maps from compact domains.
method Established second variation formula of bienergy, examined stability of previously constructed maps.
result Existence of an explicit family of stable proper biharmonic maps to Euclidean spheres.
Proper actions on bornological spaces are characterized with compatible coarse structures.
problem Characterizing proper actions on bornological spaces.
method Proving the existence of compatible coarse structures for proper actions.
result Bornological spaces admit compatible coarse structures for proper actions.
Reduces constructing multiplicative connections to simpler tasks.
problem Constructing multiplicative connections on proper Lie groupoids.
method Reduction to simpler tasks involving proper and regular Lie groupoids.
result Simpler methods for constructing multiplicative connections.
New method constructs proper affine actions of groups in higher dimensions.
problem Finding proper affine actions of discrete groups in higher-dimensional spaces.
method Higher strip deformations and Margulis invariant for properness.
result Affine actions of convex cocompact groups and virtually free groups are constructed properly.
Groups with specific subgroup actions have proper actions on uniformly convex spaces.
problem Proper actions of hyperbolic relative groups on Banach spaces.
method Affine isometric actions on uniformly convex Banach spaces.
result Groups with proper actions on uniformly convex spaces.
Estimates proper calibration errors and refinement terms in probabilistic predictions.
problem Lack of a general estimator for proper calibration errors and refinement terms with known statistical properties.
method Proposes a method for consistent, asymptotically unbiased estimation of proper calibration errors and refinement terms.
result Proves the relation between refinement and f-divergences, implying information monotonicity in neural networks.
Proves existence of proper solutions for inverse mean curvature flow.
problem Existence of proper solutions for inverse mean curvature flow.
method Proves existence theorem assuming non-degeneracy conditions on isoperimetric profile.
result No curvature assumption in existence theorem.
Study on proper-biharmonic flat tori in spheres with CMC conditions.
problem Finding conditions for CMC proper-biharmonic immersions of tori in spheres.
method Analyzing rectangular and square tori, finding necessary and sufficient conditions, and explicit expressions.
result Explicit expressions of some CMC proper-biharmonic immersions of certain tori in spheres.
Study proper actions of Lie groups on symmetric spaces, finding rigidity results and Hurwitz-Radon numbers.
problem Proper actions of non-compact semisimple Lie groups on pseudo-Riemannian symmetric spaces.
method Analysis of symmetric spaces and rigidity results.
result Any connected non-compact semisimple Lie group acting properly on these spaces must be globally isomorphic to Spin(n,1) up to compact factors. Study on proper learning under relaxed worst-case robust loss for VC classes.
problem Proper adversarially robust PAC learning under relaxed worst-case robust loss.
method Introduced a family of robust loss relaxations and showed their effectiveness for proper learnability.
result VC classes are properly PAC learnable with sample complexity close to standard PAC learning setup.
We prove that every bordered Riemann surface admits a complete proper holomorphic immersion into a ball of C^2, and a complete proper holomorphic embedding into a ball of C^3.
First we show that a curvature-adapted proper complex equifocal submanifold is a principal orbit of a Hermann type action under certain condition. Next we show that a proper complex equifocal submanifold is curvature-adapted under certain condition.
In this paper, using the framework of equivariant differential geometry, we study proper SO(p+1)×SO(q+1)-invariant biconservative hypersurfaces into the Euclidean space Rn (n=p+q+2) and proper SO(p+1)-invariant biconservative hypersurfaces into the Euclidean space Rn (n=p+2). Mo…
Proper proximality proved for various groups on non-positive curvature spaces.
problem Proper proximality of groups acting on non-positive curvature spaces.
method Established proper proximality for groups acting on CAT(0) spaces and hierarchically hyperbolic groups. result Proper proximality of many groups including mapping class groups and subgroups of curve graphs.
The paper classifies maps from vector bundles to Euclidean spaces.
problem Classifying homotopy classes of maps from vector bundles to Euclidean spaces.
method Using homotopy theory and vector bundles.
result Computed homotopy classes of proper maps and stability range.
In [5] I solved the Thom's conjecture that a proper Thom map is triangulable. In this paper I drop the properness condition in the semialgebraic case and, moreover, in the definable case in an o-minimal structure.
We study proper losses for discrete generative models without knowing the target distribution.
problem Evaluating generative models in the discrete setting without direct access to the target distribution.
method Define and construct black-box proper losses using statistical estimation theory.
result Black-box proper losses must be of polynomial form and involve more samples than the polynomial degree.
Study shows only one type of proper domain in certain spaces.
problem Classifying proper domains in Hermitian symmetric spaces.
method Analyzing Shilov boundaries and automorphism groups.
result Classification of closed proper manifolds locally modeled on Shilov boundaries.
New spaces found without certain actions, using special subgroups.
problem Existence of proper actions on homogeneous spaces.
method Using convex cocompact representations and nilpotent orbits theory.
result Found new homogeneous spaces without specific actions.
Improves model calibration for deep neural networks using proper scores.
problem Calibration errors in deep neural networks are often biased and inconsistent.
method Introduces proper calibration errors related to proper scores.
result Demonstrates the superiority of proper scores over common estimators.
New minimal surfaces grow area very quickly.
problem Understanding minimal surfaces with rapid area growth.
method Examples of minimal immersions in Euclidean space.
result Proper minimal surfaces with rapid area growth found.
Proves stable properties of proper maps between manifolds.
problem Stability of homotopy classes of proper maps and Pontryagin-Thom construction.
method Explicit construction and proof of bijection in a stable range.
result Stabilization of homotopy classes of proper maps and Pontryagin-Thom type bijection.
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…
Proves the bending map is proper for hyperbolic 3-manifolds.
problem Properness of the bending map in hyperbolic 3-manifolds.
method Analyzes geometric properties and isotopy classes of homeomorphisms.
result Proving the bending map is proper for hyperbolic 3-manifolds.
Reduces proper actions to simpler core actions for analysis.
problem Understanding properties of proper actions on manifolds.
method Extending Skjelbred and Straume's construction to non-compact groups, focusing on core of actions.
result Properties of proper actions are determined by simpler core actions.
Proper branched coverings are homeomorphisms on 3D balls or when branch set is empty.
problem Global injectivity of proper branched coverings on Euclidean balls.
method Analyzing the global injectivity of proper branched coverings defined on the Euclidean n-ball. result Proper branched coverings are homeomorphisms on 3D balls or when branch set is empty.
We first introduce an invariant index for G-equivariant elliptic differential operators on a locally compact manifold M admitting a proper cocompact action of a locally compact group G. It generalizes the Kawasaki index for orbifolds to the case of proper cocompact actions. Our invariant index is used to show that an a…
We establish several characterizations of Anosov representations of word hyperbolic groups into real reductive Lie groups, in terms of a Cartan projection or Lyapunov projection of the Lie group. Using a properness criterion of Benoist and Kobayashi, we derive applications to proper actions on homogeneous spaces of red…
The paper defines a new metric and studies proper biharmonic maps on tangent bundles.
problem Investigating proper biharmonic maps on tangent bundles.
method Defined Mus-Gradient metric on tangent bundle TM, characterized proper biharmonic maps.
result Characterized a new class of proper biharmonic maps.
Paper constructs Witten and elliptic genera for noncompact manifolds with proper actions.
problem Constructing genera for noncompact manifolds with proper actions.
method Using proper cocompact actions by almost connected Lie groups, the paper constructs and proves properties of Witten and elliptic genera.
result Vanishing and rigidity results generalizing known compact group actions.