Abstract relates Lipschitz-Killing measures to polar volumes of definable sets.
problem Relating geometric measures of definable sets to their polar images.
method Relates Lipschitz-Killing measures to volumes of generic polar images for smooth submanifolds, extending to infinitesimal versions.
result Establishes a relation between polar invariants and densities of generic polar images.
Curvature measures uniquely determined by invariance under embeddings.
problem Characterizing curvature measures uniquely.
method Applied Weyl principle and Künneth-type formula.
result Curvature measures uniquely characterized by invariance under isometric embeddings.
The abstract extends curvature measures to pseudo-Riemannian manifolds.
problem Extending curvature measures to pseudo-Riemannian manifolds.
method Constructing a family of generalized curvature measures.
result Generalized curvature measures behave naturally under isometric immersions.
Paper classifies curvature measures and confirms a conjecture.
problem Classifying curvature measures and proving the angularity conjecture.
method Investigation of translation-invariant angular curvature measures and use of isometric immersions and Lipschitz-Killing algebra.
result Confirmation of the angularity conjecture.
Fractal Lipschitz-Killing curvature measures C^f_k(F,.), k = 0, ..., d, are determined for a large class of self-similar sets F in R^d. They arise as weak limits of the appropriately rescaled classical Lipschitz-Killing curvature measures C_k(F_r,.) from geometric measure theory of parallel sets F_r for small distances…
Applying a local Gauss-Bonnet formula for closed subanalytic sets to the complex analytic case, we obtain characterizations of the Euler obstruction of a complex analytic germ in terms of the Lipschitz-Killing curvatures and the Chern forms of its regular part. We also prove analogous results for the global Euler obstr…
We prove a formula that relates the Euler-Poincaré characteristic of a closed semi-algebraic set to its Lipschitz-Killing curvatures
New valuations on contact manifolds generalize Euclidean integral geometry.
problem Understanding curvature in contact geometry.
method Reinterpreting valuations from Riemannian to contact manifolds.
result Contact manifolds have canonical families of generalized valuations.
The Weyl principle holds in some Finsler settings despite general failure.
problem Applying the Weyl principle to Finsler manifolds.
method Investigation of the Weyl principle in Finsler geometry.
result A weak form of the Weyl principle persists in certain Finsler settings.
Paper proves total curvature for convex hypersurfaces in equiaffine space.
problem Understanding total curvature for equiaffine immersions.
method Analyzes the equality case of Lipschitz--Killing curvature inequality.
result Total absolute curvature equals 2 for convex hypersurfaces.
Study on random manifolds converging to intrinsic geometry.
problem Understanding the intrinsic geometry of random manifolds.
method Sequence of random embeddings into Euclidean spaces, convergence analysis.
result Many intrinsic functionals converge to deterministic limits.
Study of cosmic microwave background polarization using spin random fields.
problem Detecting deviations from Gaussianity and anisotropies in cosmic fields.
method Explicit formula for Lipschitz-Killing curvatures of spin spherical random fields.
result Coherent with asymptotic results, providing new metric expressions.
We show how Alesker's theory of valuations on manifolds gives rise to an algebraic picture of the integral geometry of any Riemannian isotropic space. We then apply this method to give a thorough account of the integral geometry of the complex space forms, i.e. complex projective space, complex hyperbolic space and com…
Developed new Crofton formulas for pseudo-Riemannian spaces.
problem Computing volumes and curvature integrals in pseudo-Riemannian space forms.
method Introduced Crofton formulas using distributions and Alesker's Radon transform.
result Explicit Crofton formulas for all isometry-invariant valuations on pseudo-Riemannian spaces.
Study on spin random fields using chaos decomposition for cosmic microwave background modeling.
problem Modeling polarization of Cosmic Microwave Background using spin random fields.
method Explicit Wiener-Itô chaos decomposition of area measures of level sets.
result Reveals a clear difference between high frequency regime and zero spin case.
Let M be a differentiable manifold. We say that a tensor field g defined on M is non-regular if g is in some local Lp space or if g is continuous. In this work we define a mollifier smoothing g_t of g that has the following feature: If g is a Riemannian metric of class C2, then the Levi-Civita connection and the Rieman…
The paper characterizes the geometry and topology of spin random fields.
problem Understanding the expected geometry and topology of spin random fields.
method Investigating the asymptotic behavior of geometric and topological functionals for spin random fields under scaling assumptions.
result Explicit results for monochromatic fields, showing non-universal asymptotic behavior and new generalized models.
Thurston's circle packing approximation of the Riemann Mapping (proven to give the Riemann Mapping in the limit by Rodin-Sullivan) is largely based on the theorem that any topological disk with a circle packing metric can be deformed into a circle packing metric in the disk with boundary circles internally tangent to t…
For germs of subanalytic sets, we define two finite sequences of new numerical invariants. The first one is obtained by localizing the classical Lipschitz-Killing curvatures, the second one is the real analogue of the evanescent characteristics introduced by M. Kashiwara. We show that each invariant of one sequence is …
GeoTop resolves topological ambiguity in diagnostic imaging using geometric-topological analysis.
problem Topological equivalence between benign and malignant structures in diagnostic images.
method Combines Topological Data Analysis and Lipschitz-Killing Curvatures to resolve ambiguity.
result Achieves 3.6% accuracy improvement and reduces false positives/negatives by 15-18%.
We study the differential geometric consequences of our previous result on the existence of fat triangulations, in conjunction with a result of Cheeger, Müller and Schrader, regarding the convergence of Lipschitz-Killing curvatures of piecewise-flat approximations of smooth Riemannian manifolds. A further application t…
Investigate the local geometry of smooth surfaces in 4-space via contact with 2-planes and apparent contours.
problem Local geometry of smooth surfaces in 4-space
method Contact with 2-planes and apparent contour
result Prove connections between singularities of parallel projections, orthogonal projections, and height functions.
We consider vector valued, unit variance Gaussian processes defined over stratified manifolds and the geometry of their excursion sets. In particular, we develop an explicit formula for the expectation of all the Lipschitz--Killing curvatures of these sets. Whereas our motivation is primarily probabilistic, with statis…
Weyl's intrinsic volumes converge to the Euler characteristic of the base manifold under certain metrics.
problem Convergence of intrinsic volumes on Riemannian manifolds.
method Defined a new metric and used it to study the convergence of intrinsic volumes.
result Intrinsic volumes converge to the Euler characteristic of the base manifold.
The paper studies residues of manifolds and their applications in geometry.
problem Understanding the residues of manifolds and their geometric implications.
method Analytic continuation and Möbius invariance of residues, introduction of relative and weighted residues.
result Scalar curvature, mean curvature, and Euler characteristic can be expressed in terms of residues.
Study invariant measures on measured laminations for subgroups of mapping class group.
problem Classify invariant Radon measures on space of measured laminations for subgroups of mapping class group.
method Geometric approach, focusing on recurrent measured laminations, explicitly constructing ergodic measures.
result Show uniquely ergodic for divergence-type subgroups, generalize results for full mapping class group.
New set-valued star-shaped risk measures introduced for better risk assessment.
problem Improving risk assessment in financial contexts.
method Developed new set-valued star-shaped risk measures and proved their representation theorems.
result Set-valued star-shaped risk measures can be represented as unions of set-valued convex risk measures.
The Bergman measure converges to the Zhang measure on a hybrid space.
problem Proving convergence of Bergman measures to Zhang measure.
method Analyzing convergence on a hybrid space and metrized curve complex.
result Bergman measure converges to Zhang measure on a hybrid space.
Bayesian approach to robust risk measures under model uncertainty.
problem Representing robust risk measures as a single probability measure.
method Introducing two types of risk measures and analyzing their relation to robust risk measures.
result Robust risk measures can be represented by a mixture probability measure, a Bayesian approach.
Survey of time consistency in dynamic risk and performance measures using LM-measure.
problem Time consistency of dynamic risk and performance measures in discrete time.
method Focus on LM-measure and update rule to study time consistency.
result Comprehensive overview of time consistency properties.
The paper classifies 1-dimensional uniform measures in various dimensions.
problem Classifying uniformly distributed measures of dimension 1 in general codimension.
method Analyzing measures with connected 1-dimensional support and providing a partial classification for general measures.
result Uniform measures with connected 1-dimensional support are homogeneous measures.
The paper studies dynamic star-shaped risk measures and their representation.
problem Representing dynamic star-shaped risk measures and their properties.
method Representation theorems for dynamic monetary and star-shaped risk measures.
result Dynamic star-shaped risk measures can be represented as the lower envelope of a family of dynamic convex risk measures.
Introduces Star-Shaped deviation measures for risk analysis.
problem Risk measurement and analysis in finance.
method Characterizes Star-Shaped deviation measures through acceptance sets and convex deviation measures.
result Exposes the relationship between Star-Shaped risk measures and deviation measures.
Transformers can interpolate between arbitrary measures.
problem Understanding the expressive power of Transformers as measure-to-measure maps.
method Provided an explicit choice of parameters for a single Transformer to match N arbitrary input measures to N arbitrary target measures.
result A single Transformer can interpolate between arbitrary measures.
Classifies invariant measures on specific character varieties.
problem Classifying invariant probability measures on character varieties.
method Measure disintegration along transverse Lagrangian tori fibrations.
result Ergodic measures are either counting measures on finite orbits or Liouville measures.
Researchers compute the ratio between two normalizations of Thurston measure on measured laminations.
problem Computing the ratio between two normalizations of Thurston measure.
method Using the integral and symplectic structures on the space of measured laminations.
result Computed the ratio between two normalizations of Thurston measure.
Paper characterizes star-shaped risk measures and their properties.
problem Characterizing risk measures in the presence of liquidity risk and competitive delegation.
method Characterization of star-shaped risk measures, study of their properties.
result Star-shaped risk measures include all practically used risk measures.
Theorem generalizes Reifenberg's for measures with bounds on β-numbers.
problem Bounding measures away from k-rectifiable sets with β-numbers.
method Assumptions on Jones' β-numbers to measure closeness to subspaces.
result Effective measure bounds on μ away from a closed k-rectifiable set.
New measures found in 3-uniform geometry.
problem Understanding non-flat uniform measures in geometric measure theory.
method Combining combinatorial methods and distance symmetry properties.
result Infinite family of 3-uniform measures constructed.
Paper compares fairness measures and feature importance measures using SHAP.
problem Comparing fairness measures and feature importance measures.
method Focus on SHAP, a game-theoretic measure of feature importance.
result Results for unfairness-prone datasets.
A new method calculates a barycenter for probability measures using Wasserstein distance.
problem Finding a central measure for a set of probability distributions.
method Regularizing the pushforward measure of a set of probability distributions into the Wasserstein space and then finding the barycenter.
result The method yields a uniquely defined barycenter measure supported on the barycentric points of the input measures.
New Bayesian method for spectral deconvolution with Poisson noise.
problem Estimating physical model parameters from noisy spectral data.
method Bayesian measurement framework applied to Poisson noise model.
result Clarifies relationship between measurement time and estimation limits.
Paper introduces quasi-logconvex risk measures and their properties.
problem Characterizing and understanding new risk measures.
method Characterization through dual representation and properties of acceptance sets.
result Established dual representation and taxonomy of quasi-logconvex risk measures.
Submodularity is studied for convex risk measures, including Expected Shortfall.
problem Characterizing submodularity in convex risk measures.
method Analyzing submodularity properties of law-invariant coherent risk measures, including Expected Shortfall and Value-at-Risk.
result AES is submodular only when it reduces to ES, and empirical analysis shows AES violations are less frequent than VaR and ES violations.
Paper compares graph and set partition measures for graph clustering.
problem Comparing graph clustering methods using different similarity measures.
method Introduces graph-aware partition similarity measures and compares them with set partition measures.
result Graph-aware measures provide complementary information to set partition measures.
New geometric measure simplifies complex analysis.
problem Complex geometric analysis challenges.
method Geometric integration and convergence methods.
result Smallest measure satisfying Area Formula.
The paper explores non-convex risk measures and their characterizations.
problem Characterizing non-convex risk measures without convexity or weak convexity.
method Characterizes monetary risk measures as lower envelopes of families of convex or coherent risk measures, considering law-invariance and SSD-consistency.
result Unified representation theorems for law-invariant risk measures, including VaR.
The paper calculates extreme measures in continuous time conic finance.
problem Determining valuation bounds for financial claims.
method Using dynamic spectral risk measures and estimating extreme measures from market data.
result Explicit formulas for extreme measures' Radon-Nykodim derivatives and estimation methods.