Combines topological and geometric approaches to data analysis.
problem Understanding when and how geometric objects intersect.
method Connects topological and geometric concepts of curvature.
result Reconceptualizes curvature and links it to hyperconvexity.
Study geometric flows with varying parameters and prove continuous dependence.
problem Continuous dependence of flows on parameters in geometric settings.
method Derived suitable topologies for vector fields and flows, proved new continuous dependence.
result Proved continuous dependence of flows on parameters in a general topological space.
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%.
Study topological quantum mechanics on orbifolds with geometric interpretation.
problem Quantum mechanical models on symplectic orbifolds.
method Explicit orbifold version of quantum HKR map and exact semi-classical approximation.
result Geometric and quantum field theoretic interpretation of orbifold algebraic index.
Study on random surfaces in hyperbolic 3-manifolds, focusing on geometric and topological properties.
problem Distribution of nearly geodesic surfaces in hyperbolic 3-manifolds.
method Invariant measures on the Grassmann bundle G(M) derived from limits of random minimal surfaces.
result Topological limiting measures are totally scarring if M contains a totally geodesic subsurface, while geometrical limiting measures are not.
Topological parallax assesses AI models' geometric similarity to datasets for safety.
problem Ensuring AI models' robustness and safety in deep learning applications.
method Topological parallax compares a trained model to a reference dataset using Rips complexes and geodesic distortions.
result Topological parallax indicates whether a model shares similar multiscale geometric features with the dataset.
Introduces sheaf quantization, a topological approach to geometric quantization.
problem Topological realization of WKB-states in geometric quantization.
method Enhancement of constructible sheaves, Betti counterpart of Fukaya--Floer theory.
result Introduction to sheaf quantization as a topological realization of WKB-states.
This paper proposes a new method for learning covers of geometric datasets to improve topological inference and visualization.
problem Improving topological inference and visualization of large-scale geometric datasets.
method Proposes a method for learning topologically-faithful covers of geometric datasets using optimization.
result Simplicial complexes obtained from learned covers outperform standard methods in terms of size and representation of large-scale topology.
Survey on geometric, analytic, and topological aspects of 4D equations.
problem No specific problem stated in abstract.
method Geometric, analytic, and topological discussions.
result New solution of the Cauchy problem over null hypersurfaces.
Study the structure of equidistant decompositions in manifolds.
problem Geometric and topological structure of equidistant decompositions.
method Investigation of Riemannian manifolds.
result Detailed understanding of equidistant decompositions.
The paper explores the geometry and topology of DNN decision boundaries.
problem Understanding the geometric and topological properties of DNN decision boundaries.
method Differential geometry and the Gauss-Bonnet-Chern theorem.
result Computed the Euler characteristics of compact decision boundaries.
Formulates a new connection between topological and geometric categories.
problem No specific problem stated; focuses on category formulation.
method Formulates a connection between a topological and geometric category.
result Provides a more precise and improved version of a previous proposal.
GeoPhy uses geometric gradients to efficiently infer phylogenetic trees from molecular data.
problem Challenges in accurately inferring species relationships from molecular data due to combinatorially vast tree topologies.
method Introduces a novel, fully differentiable formulation of phylogenetic inference using geometric spaces and variational Bayesian methods.
result Significantly outperforms other approximate Bayesian methods in inferring phylogenetic trees.
The paper proposes conjectures about moduli space rigidity.
problem Rigidity of moduli spaces in algebraic geometry.
method Group-theoretic, topological, and holomorphic approaches.
result Some conjectures have been proved.
Geometric techniques reveal new insights into Gromov-Witten invariants.
problem Formulating Gromov-Witten invariants for complete intersections in projective space.
method Combining geometric group theory and geometric topology, focusing on geodesic laminations.
result Primitive cohomologies unify mathematical formulations of Gromov-Witten invariants.
The space of n-sided polygons embedded in three-space consists of a smooth manifold in which points correspond to piecewise linear or ``geometric'' knots, while paths correspond to isotopies which preserve the geometric structure of these knots. The topology of these spaces for the case n = 6 and n = 7 is described. In…
New method identifies vanishing arcs for curve singularities.
problem Characterizing arcs sent to geometric vanishing cycles.
method Introducing geometric variation operator and vanishing arcsets.
result Existence of topological exceptional collections of arcsets.
A new geometric framework resolves singularities in anomalous transport.
problem Mathematical singularities in quantum Berry connections.
method Hodge-de Rham decomposition of the Brillouin zone.
result A smooth geometric proxy potential for anomalous transport.
Neural nets learn robust geometric data representations.
problem Ensuring neural networks are robust to adversarial attacks.
method Topological Data Analysis via persistence diagrams, Lipschitz stability.
result Certified ε-robustness on ORBIT5K dataset. Flexible approach for normal approximations in geometric and topological statistics.
problem Normal approximation for complex statistics not expressible as sums of score functions.
method Flexible add-one cost operator combined with strong stabilization theory.
result Established normal approximation results for geometric and topological statistics.
JORC-UMAP improves UMAP by incorporating geometric and topological priors.
problem UMAP's local Euclidean distance assumption fails to capture intrinsic manifold geometry, leading to topological tearing and structural collapse.
method JORC-UMAP introduces Ollivier-Ricci curvature as a geometric prior and Jaccard similarity as a topological prior to reinforce edges and reduce redundant links.
result JORC-UMAP reduces tearing and collapse more effectively than standard UMAP and other DR methods, as measured by SVM accuracy and triplet preservation scores.
ICLR 2021 challenge in computational geometry and topology attracted 16 teams.
problem Designing and evaluating computational methods in differential geometry and topology.
method Designing and hosting an open-source competition with repositories Geomstats and Giotto-TDA.
result 16 teams participated in the challenge, showcasing innovative contributions to computational geometry and topology.
Geometric analysis on real analytic manifolds using seminorms.
problem Characterizing operations on real analytic manifolds and vector bundles.
method Using seminorms and geometric decompositions of jet bundles.
result New characterizations of real analytic mappings and operations.
Topologically and geometrically engaging actions have proved to be useful to obtain rigidity results for semisimple Lie group actions. We show that the action of a simple noncompact Lie group on a compact manifold preserving a unimodular rigid geometric structure of algebraic type (e.g. a connection together with a vol…
In this study, we present and analyze a framework for geometric and topological estimation for mapping of unknown environments. We consider agents mimicking motion behaviors of cyborg insects, known as biobots, and exploit coordinate-free local interactions among them to infer geometric and topological information abou…
Geometric proof of contractibility of unitary group in strong topology.
problem Contractibility of unitary group in strong operator topology.
method Direct geometric proof and construction of special subspaces and operators.
result Direct geometric proof of contractibility theorem.
Study on 3D surfaces and tangles formed by Poncelet triangles.
problem Geometric and topological properties of 3D surfaces and tangles.
method Exploration of Poncelet triangles and associated points.
result Properties of 3D surfaces and tangles formed by Poncelet triangles.
The study examines Eschenburg orbifolds with positive sectional curvature and their geometric/topological properties.
problem Understanding the geometric and topological constraints of positively curved Eschenburg orbifolds.
method Proved restrictions on singular sets and computed orbifold cohomology rings.
result Distinctive behavior in cohomology groups of positively curved Eschenburg orbifolds.
Expands Bredon's trick for applications in geometry and topology.
problem Local-to-global extension principles in geometric and topological contexts.
method Novel applications and frameworks for stratified pseudomanifolds, Ricci flow, and persistent homology.
result Establishes Bredon's trick as a unifying framework.
Unified geometric flows improve deep learning efficiency and simplify neural network topologies.
problem Improving deep learning performance and simplifying neural network structures.
method Proposes a thermodynamically coupled Ricci flow that dynamically adapts parameter space geometry to loss landscape topology, enabling automated singularity resolution and providing entanglement entropy bounds.
result Demonstrates 2.1× convergence acceleration and 63% topological simplification while maintaining O(NlogN) complexity, outperforming Riemannian baselines by 15.2% in few-shot accuracy. Study of foliations' geometric and topological structures.
problem Analyzing the geometric and topological properties of transversely affine foliations.
method Attach holonomy group and quotient stack, identify reparametrisations, classify them, and study the Kato-Nakayama space.
result Holonomy group controls the geometric part, while the Kato-Nakayama space captures the topological and dynamical aspects.
Generalizes sigma model with Lie algebroid structure and geometric conditions.
problem Consistency of constraints and gauge symmetry in topological sigma models.
method Analysis of geometric conditions and constraints in Hamiltonian and Lagrangian formalisms.
result Identifies universal compatibility condition between Lie algebroid and multi-symplectic structure.
This book is expository and is in Russian. It is shown how in the course of solution of interesting geometric problems (close to applications) naturally appear main notions of algebraic topology (homology groups, obstructions and invariants, characteristic classes). Thus main ideas of algebraic topology are presented w…
Geometric flows have proved to be a powerful geometric analysis tool, perhaps most notably in the study of 3-manifold topology, the differentiable sphere theorem, Hermitian-Yang-Mills connections and canonical Kaehler metrics. In the context of G_2 geometry, there are several geometric flows which arise. Each flow prov…
Mathematical pipeline identifies structural homology of knotted proteins.
problem Quantification and classification of protein structures, especially knotted proteins, require noise-free and complete data.
method Developed a geometric framework using persistent homology to analyze protein structures.
result Persistent homology accurately represents structural homology of knotted proteins and identifies geometric features of protein entanglement.
The paper explores geometric properties of free boundary hypersurfaces in balls.
problem Geometric properties of free boundary hypersurfaces in balls.
method Establishing relationships between geometric quantities of hypersurfaces and their boundaries.
result Umbilical points of free boundary surfaces in the unit ball depend only on their topology.
The study explores geometric properties of hyperbolic cohomology classes on Kähler manifolds.
problem Understanding the geometric effects of hyperbolic cohomology classes on Kähler manifolds.
method Introducing Kähler topologically hyperbolic manifolds and proving spectral gap theorems for positive holomorphic Hermitian vector bundles.
result Kähler topologically hyperbolic manifolds are not uniruled nor bimeromorphic to compact Kähler manifolds with trivial first real Chern class.
The study explores how different Grothendieck topologies and functors between categories preserve locality.
problem Exploring relationships between different Grothendieck topologies and functors.
method Using Grothendieck topologies and functors to relate categories and geometric objects.
result Objects like sheaves, groupoids, and functors are invariant under equivalences of Grothendieck topologies and certain functors.
New stable minimal hypersurfaces found in 4-manifolds, proving topology results.
problem Finding stable minimal hypersurfaces with specific topologies in 4-manifolds.
method Geometric measure theory and 4-manifold topology techniques.
result Existence of stable minimal hypersurfaces diffeomorphic to S3 or S2imesS1. New unknots with geometric constraints exist, proving a long-standing conjecture.
problem Existence of distinct isotopy classes of physical unknots with geometric constraints.
method Parametrised thickness and geometric thresholds to fragment isotopy classes.
result Existence of gordian unknots with prescribed geometric constraints.
In earlier work we introduced topologically minimal surfaces as the analogue of geometrically minimal surfaces. Here we strengthen the analogy by showing that complicated amalgamations act as barriers to low genus, topologically minimal surfaces.
In this note we provide natural optimal geometric conditions for a Riemannian manifold suitably covered by two open metric balls to be homeomorphic to a sphere. This can be viewed as a geometric analogue of Brown's theorem in topology stating that a closed manifold covered by two topological balls is a sphere.
The paper analyzes diffusion condensation for data geometry and topology.
problem Understanding the geometry and topology of high-dimensional data.
method Time-inhomogeneous diffusion process with geometric, spectral, and topological analysis.
result The condensation process defines intrinsic condensation homology and ambient persistent homology.
The paper explores how topological methods can reveal insights into electric charge distributions on knots.
problem Understanding the qualitative behavior of electric potentials on knots.
method Geometric topology techniques applied to electrostatics.
result Proved a lower bound on the size of the critical set based on knot projections.
A new method clusters complex networks using topological and geometric structure.
problem Clustering complex networks with intricate topology.
method Centroid-based clustering strategy using Wasserstein distance and barycenter for persistence barcodes.
result Demonstrated effectiveness on simulated and real-world networks.
A new method for group invariant machine learning using geometric projections.
problem Supervised group invariant and equivariant machine learning.
method Geometric topology approach involving projection of input data into a geometric space parametrizing symmetry group orbits.
result Improvement in accuracy compared to existing methods.
A Riemannian manifold is called geometrically formal if the wedge product of harmonic forms is again harmonic, which implies in the compact case that the manifold is topologically formal in the sense of rational homotopy theory. A manifold admitting a Riemannian metric of positive sectional curvature is conjectured to …
This survey paper concerns mainly with some asymptotic topological properties of finitely presented discrete groups: quasi-simple filtration (QSF), geometric simple connectivity (GSC), topological inverse-representations, and the notion of easy groups. As we will explain, these properties are central in the theory of d…