New surfaces with integrable geodesic flows are constructed.
problem Integrable geodesic flows on complex surfaces.
method Generalized tubular sub-manifolds and Jacobi fields.
result Conditions for integrable geodesic flows are derived.
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48 results for “geodesic sub-manifolds”
New method to visualize hyperbolic 4-manifolds and study mutations.
problem Understanding and visualizing hyperbolic 4-manifolds and mutations.
method Orbifold covers of Coxeter polytopes with facet colorings and studying totally geodesic sub-manifolds.
result Construction of hyperbolic 4-manifolds with unique properties.
In this paper, we introduce a concept of B-minimal sub-manifolds and discuss the stability of such a sub-manifold in a Riemannian manifold . Assume is a smooth function on . By definition, we call a sub-manifold {\em B-minimal} in if the product sub-manifold is a {\em minimal}…
Global Lipschitz geometry of conic singular sub-manifolds with applications to algebraic setsmath.DG
Conic singular sub-manifolds are Lipschitz Normally Embedded in compact non-Euclidean manifolds.
problem Understanding the Lipschitz geometry of conic singular sub-manifolds.
method Analyzing the metric properties of conic singular sub-manifolds in compact non-Euclidean manifolds.
result Connected conic singular sub-manifolds are Lipschitz Normally Embedded.
Optimal transport natural gradient improves optimization in statistical models.
problem Improving optimization in statistical models with continuous sample spaces.
method Pulling back the Wasserstein metric tensor to a parameter space, creating a Riemannian manifold.
result Natural gradient descent outperforms standard gradient descent in Wasserstein distance optimization.
KPCA-BO improves BO for high-dimensional optimization problems by learning a non-linear sub-manifold.
problem High-dimensional optimization problems where Gaussian Process regression requires too much data and computation.
method KPCA-BO embeds a non-linear sub-manifold in the search space, learning a GPR model on this sub-manifold.
result KPCA-BO outperforms vanilla BO in convergence speed, especially as dimensionality increases.
Paper proposes methods to learn sub-manifolds and estimate densities in normalizing flows.
problem Normalizing flows struggle with finding sub-manifolds in high-dimensional data.
method Introduces per-pixel penalized log-likelihood and hierarchical training approaches.
result Validated superior performance in manifold learning and density estimation.
We prove that the critical points of various energies such as the area, the Willmore energy, the frame energy for tori...etc among possibly branched immersions constrained to evolve within a smooth sub-manifold of the Teichmüller space satisfy the corresponding constrained Euler Lagrange equation. We deduce that critic…
New convex projective structures found on non-hyperbolic 3-manifolds.
problem Understanding when convex projective structures can be applied to non-hyperbolic 3-manifolds.
method Deforming cusped hyperbolic 3-manifolds to convex projective structures with totally geodesic boundaries and gluing them.
result Many doubles of cusped hyperbolic 3-manifolds admit convex projective structures.
We define an integer graded symplectic Floer cohomology and a Fintushel-Stern type spectral sequence which are new invariants for monotone Lagrangian sub-manifolds and exact isotopes. The Z-graded symplectic Floer cohomology is an integral lifting of the usual Z_Sigma(L)-graded Floer-Oh cohomology. We prove the Kunneth…
The paper studies Finsler spheres with constant flag curvature and finite orbits of prime closed geodesics.
problem Investigating Finsler spheres with specific curvature properties and geodesic orbits.
method Analyzing the action of isometries and loops on Finsler spheres, focusing on finite orbits of prime closed geodesics.
result The existence of geometrically distinct orbits of prime closed geodesics and their properties.
Inference for normal and Monte Carlo distributions using minimum relative entropy.
problem Inference from partial information on expectations and covariances.
method Minimum relative entropy sub-manifolds, analytical formulas, Monte Carlo simulations.
result Improved numerical implementation for inference from partial information.
I-BBS identifies latent sub-manifolds from distance matrices, robust to noise.
problem Identifying latent sub-manifolds from distance matrices in high-dimensional spaces.
method Coordinate-free inference using random distance matrix theory and generative noise models.
result Recovering latent geometry from integer-stable signatures of eigenvalues.
Many procedures in science, engineering and medicine produce data in the form of geometric shapes. Mathematically, a shape can be modeled as an un-parameterized immersed sub-manifold, which is the notion of shape used here. Endowing shape space with a Riemannian metric opens up the world of Riemannian differential geom…
We consider the -dimensional Euclidean space, , with certain -dimensional compact, closed and orientable sub-manifolds (which we call \emph{singularity manifolds} and represent by ) removed from it. We define and investigate the problem of finding a homotopy-like class i…
Paper develops geometric tools for power flow analysis.
problem Analyzing the boundary of power flow solution space.
method Differential geometric analysis of power flow solution space boundary.
result New high precision continuation method and geodesic coordinates.
A new classifier updates sequentially using maximum margin principles.
problem Sequential data collection and partial labeling.
method Maximum margin classifier with Maximum Entropy Discrimination principle, kernel representation, and regularization.
result Improved performance compared to non-sequential classifiers.
Detects phase transitions in collective behavior using manifold curvature.
problem Identifying phase transitions in multi-agent systems.
method Uses curvature and singular value ratios to detect phase transitions on manifolds.
result Phase transitions can be detected and separated into distinct sub-manifolds.
Study smooth mean curvature flow for arbitrary co-dimension sets with weak tangent planes.
problem Existence and uniqueness of smooth mean curvature flow from Reifenberg flat sets.
method Viscosity solutions and level set approach for arbitrary co-dimension.
result Non-fattening and smooth flow for small Reifenberg parameter.
This paper introduces the concept of functional current as a mathematical framework to represent and treat functional shapes, i.e. sub-manifold supported signals. It is motivated by the growing occurrence, in medical imaging and computational anatomy, of what can be described as geometrico-functional data, that is a da…
Submanifolds of finite type were introduced by the author during the late 1970s. The first results on this subject had been collected in author's book [Total mean curvature and sub manifolds of finite type, World Scientific, NJ, 1984]. A list of ten open problems and three conjectures on submanifolds of finite type was…
We define an integer graded symplectic Floer cohomology and a spectral sequence which are new invariants for monotone Lagrangian sub-manifolds and exact isotopies. Such an integer graded Floer cohomology is an integral lifting of the usual Floer-Oh cohomology with $Z_{\Si (L)}$ grading. As one of applications of the sp…
Hyperbolic 3-manifolds can be approximated by removing Cantor sets from the 3-sphere.
problem Approximating hyperbolic 3-manifolds using Cantor set complements in the 3-sphere.
method Using exhaustion by -injective sub-manifolds and removing Cantor sets.
result Hyperbolic 3-manifolds can be geometrically approximated by removing Cantor sets from the 3-sphere.
It was recently shown that there exists an explicit bound for the number of Pachner moves needed to connect any two triangulation of any Haken 3-manifold which contains no fibred sub-manifolds as strongly simple pieces of its JSJ-decomposition. In this paper we prove a generalisation of that result to all knot compleme…
Study conic singular manifolds, proving Lipschitz normal embedding.
problem Understanding metric properties of conic singular manifolds.
method Analyzing interplay between conic and asymptotically conic behavior.
result Proves Lipschitz normal embedding for conic singular sub-manifolds.
CLAMP uses neural manifold packing to improve self-supervised learning.
problem Improving self-supervised learning for vision tasks.
method CLAMP recasts representation learning as a manifold packing problem, introducing a loss function inspired by particle systems.
result CLAMP achieves competitive performance with state-of-the-art models and separates neural manifolds effectively.
Establishes regularity of conformal harmonic maps.
problem Ensuring smoothness of harmonic maps.
method Viscosity techniques for stationary varifolds.
result Weakly conformal maps are smooth if stationary.
Study moduli spaces of solutions to Bogomolny equations on surfaces.
problem Understanding solutions to Bogomolny equations on surfaces with specific boundary conditions.
method Refined Kobayashi-Hitchin correspondence and identification with Higgs bundles.
result Identified moduli spaces as holomorphic lagrangian sub-manifolds.
In this work we consider a question in the calculus of variations motivated by riemannian geometry, the isoperimetric problem. We show that solutions to the isoperimetric problem, close in the flat norm to a smooth submanifold, are themselves smooth and -close to the given sub manifold. We show also a version …
Local functions prove cotangent paths coisotropic, avoiding Banach manifold issues.
problem Proving Poisson bivector fields without Banach manifold constraints.
method Using local functions on path space, avoiding infinite-dimensional issues.
result Cotangent paths form a coisotropic set for local functions.
Suppose that we observe noisy linear measurements of an unknown signal that can be modeled as the sum of two component signals, each of which arises from a nonlinear sub-manifold of a high dimensional ambient space. We introduce SPIN, a first order projected gradient method to recover the signal components. Despite the…
The paper generalizes results on biconservative hypersurfaces in Euclidean spaces.
problem Characterizing biconservative hypersurfaces in Euclidean spaces.
method Analyzing hypersurfaces with specific divergence properties and curvature conditions.
result Biconservative hypersurfaces with constant scalar curvature are of constant mean curvature.
New method adapts neural networks without losing prior knowledge.
problem Understanding and enabling flexible adaptation of neural networks.
method Differential geometry framework, functionally invariant paths (FIP).
result Achieves comparable state-of-the-art performance on continual learning and sparsification tasks.
Study torus knots in lens spaces using Gromov-Witten invariants and topological recursion.
problem Computing Gromov-Witten invariants for torus knots in lens spaces.
method Construct Lagrangian sub-manifolds and relate to topological recursion.
result Verify a conjecture in lens space for Gromov-Witten invariants.
New metrics differentiate effective OOD sets for training calibrated CNNs.
problem Vanilla CNNs struggle with out-of-distribution (OOD) samples.
method Developed metrics based on generalization errors of Augmented-CNN.
result Most protective OOD sets lead to better A-CNN performance.
The study proves lower semi-continuity of the index for minimal surfaces using viscosity methods.
problem Proving lower semi-continuity of the index for minimal surfaces.
method Using Hilbert manifold structure and viscous approximation of the area.
result Proves lower semi-continuity of the index for minimal surfaces.
The geometry of oscillatory integrals on manifolds with intermediate symmetry.
problem Classification of curvature conditions in Sogge's program.
method Proposing a classification of curvature conditions.
result No manifolds satisfy the chaotic curvature condition of order 1.
A new method simulates implied volatility surfaces for multiple assets.
problem Generating consistent market scenarios for multiple asset implied volatilities.
method Combining functional data analysis and neural SDEs with a penalty for model misspecification.
result Simulated market scenarios are consistent with historical features and lie within the sub-manifold of essentially free static arbitrage.
New geometric properties discovered in a specific Frobenius manifold.
problem Exploring hidden geometric aspects of a specific Frobenius manifold.
method Proved the manifold is pseudo-elliptic, sub-manifold of a Lorentzian projective manifold, and unraveled Maurer-Cartan structures.
result Found causality conditions bridging Lorentzian and probabilistic concepts.
Normalizing flows adapted for Riemannian manifolds.
problem Density estimation on Riemannian manifolds.
method Adapting normalizing flows to Riemannian geometry.
result Scalable and simple method for density estimation on manifolds.
A new multi-scale vector quantization method for unsupervised data.
problem Efficiently reconstructing unsupervised data with minimal distortion.
method Reconstruction trees, inspired by decision trees, explore data in a multi-scale fashion.
result Analysis of expected distortion under fixed unknown distribution, with asymptotic and finite sample results.
The structure of a diffeomorphism invariant Lagrangians for an extended object W embedded in a bulk space M is discussed by following a close analogy with the relativistic particle in electromagnetic field as a system that is reparametrization-invariant. The current construction naturally contains, relativistic point p…
Study uniform convergence of random walk Laplacians to diffusion Laplacian on smooth manifolds.
problem Uniform convergence of random walk Laplacians to diffusion Laplacian on smooth manifolds.
method Analysis of random walks on geometric and directed kNN graphs, using concentration tools and differential geometry.
result Uniform convergence of NN Laplacians to diffusion Laplacian, without continuity of transition kernel.
Model learns tool affordances from vision, enabling tool selection.
problem Learning tool affordances from visual input.
method Vision-based generative model with task predictor.
result Agents can select appropriate tools based on task success criteria.
Joint training with autoencoder improves relation learning in knowledge bases.
problem Learning relations in knowledge bases is challenging due to compositional constraints.
method Joint training of relations with an autoencoder to capture compositional constraints.
result Joint training leads to interpretable sparse codings and improved performance.
The paper analyzes the latent geometry of generative diffusion models.
problem The manifold overfitting phenomenon in generative models.
method Statistical physics approach to analyze the spectrum of eigenvalues and singular values of the Jacobian of the score function.
result Three distinct qualitative phases during the generative process: trivial, manifold coverage, and consolidation phases.
We write the Euler characteristic X(G) of a four dimensional finite simple geometric graph G=(V,E) in terms of the Euler characteristic X(G(w)) of two-dimensional geometric subgraphs G(w). The Euler curvature K(x) of a four dimensional graph satisfying the Gauss-Bonnet relation sum_x K(x) = X(G) can so be rewritten as …
Round spheres are uniquely characterized by half-geodesics.
problem Characterizing round spheres in Riemannian geometry.
method Establishing that Riemannian spheres with specific geodesic properties are round.
result Riemannian spheres with all geodesics closed and many half-geodesics are round.