Enhanced bounds on rho-invariants for 3-manifolds.
problem Establishing bounds on Cheeger-Gromov rho-invariants for 3-manifolds.
method Constructing chain null-homotopies with linear complexity.
result Linearly bounded complexity of constructed null-homotopies.
New method uses iterated integrals to bridge geometric and homotopy information.
problem Lack of effective methods to connect geometric and homotopy information.
method Introducing Chen's iterated integrals on loop spaces.
result Upper bounds for Gromov's distortion and non-existence of small-volume cycles.
Study connects manifold complexity to scalar curvature bounds.
problem Understanding the relationship between manifold complexity and scalar curvature.
method Combining quantitative operator K-theory, Lipschitz topological K-theory, and a vanishing theorem.
result Established a relationship between covering complexity and scalar curvature bounds.
We obtain a quantitative estimate on the generalised index of translators for the mean curvature flow with bounded norm of the second fundamental form. The estimate involves the dimension of the space of weighted square integrable f-harmonic 1-forms. By the adaptation to the weighted setting of Li-Tam theory developed …
Stable solution found for manifold topology from boundary data.
problem Determining manifold properties from boundary data and eigenvalues.
method Quantitative stability estimates and unique continuation for the wave operator.
result Eigenvalues and boundary values determine a metric space close to the manifold.
New principles prove precompactness of domains with lower Ricci curvature bound.
problem Proving precompactness of domains with lower Ricci curvature bound.
method Quantitative Hopf-Rinow theorem and doubling property.
result New precompactness principles applicable to incomplete Riemannian manifolds.
Study quantifies properties of PMC hypersurfaces with area bounds.
problem Understanding the topology and singular set of PMC hypersurfaces.
method Established quantitative topological and singularity properties for PMC hypersurfaces.
result Quantitative bounds on Betti numbers and Minkowski content of singular sets.
EKH adds metrics to knot theory, enabling more detailed analysis.
problem Lack of quantitative data in knot theory.
method Integrates metric into knot theory with evolutionary Khovanov homology (EKH).
result EKH reveals non-trivial knot invariants at appropriate scales.
Paper shows stability of metric reconstruction for orbifolds from spectral data.
problem Determining the metric structure of collapsing orbifolds from spectral data.
method Improved quantitative unique continuation for wave operator on Riemannian manifolds.
result Quantitative stability of inverse problem for Riemannian orbifolds.
Graph Neural Networks (GNNs) have achieved promising performance on a wide range of graph-based tasks. Despite their success, one severe limitation of GNNs is the over-smoothing issue (indistinguishable representations of nodes in different classes). In this work, we present a systematic and quantitative study on the o…
New non-perturbative counterexamples to Min-Oo's Conjecture are created.
problem Min-Oo's Conjecture on positive curvature and mass.
method Quantitative Gromov-Lawson Schoen-Yau surgery.
result Construction of new non-perturbative counterexamples with more complex topology.
Sharp stability result for maps near infinitely concentrated minimisers.
problem Stability of maps near minimisers with infinite concentration.
method Dynamic approach to deform maps into harmonic maps, controlling topology changes.
result Sharp quantitative estimates on map distance to infinitely concentrated minimisers.
Study the topology of loops of contactomorphisms and Legendrians in non-orderable manifolds.
problem Global topology of loops of contactomorphisms and Legendrians in non-orderable manifolds.
method Filtering loops by positivity and analyzing subspaces of the filtration.
result Homotopy groups of the space of loops are subgroups of the positive loops subspace.
We implement methods from computational homology to obtain a topological signal of singularity formation in a selection of geometries evolved numerically by Ricci flow. Our approach, based on persistent homology, produces precise, quantitative measures describing the behavior of an entire collection of data across a di…
The paper explores the topology of polygonal meshes and their properties.
problem Understanding the topological properties of polygonal meshes.
method Overview of topological concepts, definitions of intrinsic and extrinsic topology, proofs of Euler and Euler-Poincaré formulas, and discussion on cutting meshes.
result Detailed understanding and definitions of polygonal mesh topology, including intrinsic and extrinsic properties.
The paper examines how closed curves on surfaces intersect and how this intersection determines the curves.
problem Determining closed curves on surfaces based on their intersections.
method Constructing and studying k-equivalent curves, analyzing intersections with other curves. result Curves are determined by their intersections with all other curves, but non-simple curves require infinitely many intersections to distinguish.
We give soft, quantitatively optimal extensions of the classical Sphere Theorem, Wilking's connectivity principle and Frankel's Theorem to the context of k-th Ricci curvature. The hypotheses are soft in the sense that they are satisfied on sets of metrics that are open in the C2-topology.
Study shows stability of Schrödinger operator spectral data on a manifold.
problem Determining a manifold and potential function from spectral data.
method Approximation of spectral data on a subset to determine manifold and potential.
result Quantitative stability estimate for Schrödinger operator inverse problem.
The paper proves bounds on curvature and injectivity radius for convex sums of Riemannian metrics.
problem Understanding the geometry of convex sums of Riemannian metrics.
method Quantitative inverse function theorem and Riemannian geometry techniques.
result Injectivity radii of convex sums have uniform lower bounds.
Study finds topological restrictions for stable free boundary CMC surfaces in negatively curved settings.
problem Understanding topological constraints for stable free boundary CMC surfaces in negatively curved settings.
method Established intrinsic area-length-topology inequalities via a conformal upper bound for a constrained first Robin eigenvalue of the Jacobi operator.
result Explicit topological restrictions for stable free boundary CMC surfaces, showing low genus and few boundary components.
Study new Ricci flow invariant curvature conditions.
problem Topology of manifolds with pinched curvature.
method Provide quantitative evidence for an unpublished conjecture.
result Topology of manifolds with pinched curvature studied.
This study solves a Dirichlet problem for specific elliptic equations on Riemannian manifolds with concave boundaries.
problem Solving the Dirichlet problem for degenerate elliptic equations on Riemannian manifolds with mean concave boundaries.
method The proof relies on a quantitative boundary estimate.
result Analogous results are obtained in complex variables and on certain product manifolds.
This survey summarizes the results discussed in a talk at "Bielefeld Geometry & Topology Days" held at Bielefeld University in July 2015. We are interested in quantitative and qualitative properties of Bott-Chern cohomology. We announce new results obtained in [D. Angella, N. Tardini, Quantitative and qualitative cohom…
Topological method detects Hopf bifurcations from time series.
problem Detecting Hopf bifurcations in nonlinear systems from time series data.
method Persistent homology applied to Takens embedding for phase space reconstructions.
result A simple scalar topological functional identifies critical bifurcation points.
Sharp Talenti-type comparison theorem for p-Laplacian on RCD(K,N) spaces.
problem Understanding the p-Laplacian on RCD(K,N) spaces.
method Proving a Talenti-type comparison theorem.
result Sharp, rigid and stable Talenti-type comparison theorem.
Paper proves rigidity of certain Ricci shrinkers.
problem Rigidity of Ricci shrinkers in specific spaces.
method Quantitative characterization, rigidity inequality, contraction and extension.
result Uniqueness of tangent flow for compact Ricci flows.
Assessing generative models is not an easy task. Generative models should synthesize graphs which are not replicates of real networks but show topological features similar to real graphs. We introduce an approach for assessing graph generative models using graph classifiers. The inability of an established graph classi…
This work focuses on important step in quantitative topology: given homotopic mappings from Sm to Sn of Lipschitz constant L, build the (asymptotically) simplest homotopy between them (meaning having the least Lipschitz constant). The present paper resolves this problem for the first case where Hopf invariant p…
Survey on optimizing topological descriptors for machine learning.
problem Optimizing topological priors in machine learning models.
method Minimizing topologically-informed losses using gradient descent.
result Various techniques enable optimization of persistence-based loss functions.
Let M be a compact n-manifold of RicM≥(n−1)H (H is a constant). We are concerned with the following space form rigidity: M is isometric to a space form of constant curvature H under either of the following conditions: (i) There is ρ>0 such that for any x∈M, the open ρ-ball at $x^…
The holonomic approximation lemma of Eliashberg and Mishachev is a powerful tool in the philosophy of the h−principle. By carefully keeping track of the quantitative geometry behind the holonomic approximation process, we establish several refinements of this lemma. Gromov's idea from convex integration of working on…
The paper studies constraint maps with singularities and free boundaries, proving continuity near singularities and optimality.
problem Analyzing the structure of constraint maps with singularities and free boundaries.
method Establish continuity near singularities using a new quantitative unique continuation principle, and investigate the presence of branch points leading to new singularities.
result Topological singularities can only lie in the interior of the contact set in the uniformly convex setting, and the optimality of this result is proven.
In this paper, we exploit minimal sensing information gathered from biologically inspired sensor networks to perform exploration and mapping in an unknown environment. A probabilistic motion model of mobile sensing nodes, inspired by motion characteristics of cockroaches, is utilized to extract weak encounter informati…
We consider decomposition spaces R3/G that are manifold factors and admit defining sequences consisting of cubes-with-handles. Metrics on R3/G constructed via modular embeddings into Euclidean spaces promote the controlled topology to a controlled geometry. The quasisymmetric parametrizability of the metric spa…
Paper introduces a framework for diagnosing Alzheimer's disease using higher-order topological features from fMRI.
problem Diagnosing Alzheimer's disease using brain network topology.
method Persistent homology to extract higher-order features (cycles, cavities) from fMRI data.
result Framework significantly outperforms existing methods in AD classification.
The study connects projective codes to the distribution of zeros of odd maps.
problem Understanding the distribution of zeros of odd maps from spheres to Euclidean space.
method Using the topology of the space of probability measures on the sphere.
result Generalization of the Borsuk-Ulam theorem and its four consequences.
Topological Data Analysis (TDA) is a recent approach to analyze data sets from the perspective of their topological structure. Its use for time series data has been limited to the field of financial time series primarily and as a method for feature generation in machine learning applications. In this work, TDA is prese…
Introduces a Cost function to measure Legendrian knot obstructions.
problem Measuring obstructions for Legendrian knot isotopies.
method Introduces a non-negative integer-valued Cost function.
result Cost function induces a metric on topologically isotopic Legendrian knots.
CycleMorph improves image registration by preserving topology with cycle consistency.
problem Preserving original topology during deformation in image registration.
method Cycle-consistent deformable image registration approach.
result Effective and accurate registration on diverse image pairs within seconds.
Enhances topology optimization with multiclass microstructures using latent variable Gaussian process.
problem Lack of an inherent ordering or distance measure between different classes of microstructures.
method Extended latent-variable Gaussian process (LVGP) models to multi-response LVGP (MR-LVGP) models for metamaterials.
result Improved performance through consistent load-transfer paths for micro- and macro-structures.
Combines geometry and topology for analyzing hierarchical datasets.
problem Analyzing complex, hierarchical datasets with irregular structures.
method Combines manifold learning and topological data analysis.
result Superior classification results compared to state-of-the-art methods.
In this paper, we give a proof of the quantitative Morse theorem stated by {Y. Yomdin} in \cite{Y1}. The proof is based on the quantitative Sard theorem, the quantitative inverse function theorem and the quantitative Morse lemma.
This paper proves properties of uniformly hyperbolic sets and constructs Markov partitions.
problem Establishing properties of uniformly hyperbolic sets and constructing Markov partitions.
method Backward graph transform, spectral decomposition, shadowing lemma, Markov partitions construction.
result Explicit bounds and Hölder continuity for the coding map.
New spectral invariants recover Calabi invariant for surface dynamics.
problem Understanding Hamiltonian homeomorphisms and spectral invariants.
method Defining new spectral invariants for Lagrangian links in surfaces.
result Our invariants recover the Calabi invariant and resolve open questions.
Qlib aims to integrate AI into quantitative investment.
problem Challenges in applying AI to quantitative investment.
method Design and develop Qlib to accommodate AI-driven workflow.
result Qlib realizes the potential of AI technologies in quantitative investment.
While a wide range of interpretable generative procedures for graphs exist, matching observed graph topologies with such procedures and choices for its parameters remains an open problem. Devising generative models that closely reproduce real-world graphs requires domain knowledge and time-consuming simulation. While e…
We consider flat surfaces and the points of their metric completions, particularly the singularities to which the flat structure of the surface does not extend. The local behavior near a singular point x can be partially described by a topological space L(x) which captures all the ways that x can be "approached linearl…
The paper proves the existence of a continuous family of translating surfaces under a specific curvature flow.
problem Existence of convex translating surfaces under flow by α-th power of Gauss curvature. method Constructing a family of translating surfaces using Jacobi fields and analyzing their effective growth rates.
result The family of translating surfaces is a topological manifold with quantitative convergence rates.