Study homotopy motions of surfaces in 3-manifolds.
problem Understanding the behavior of surfaces under continuous deformations in 3-manifolds.
method Introduce and study homotopy motions of surfaces in closed orientable 3-manifolds.
result Systematic study of homotopy motions of surfaces in 3-manifolds.
Study the space of embeddings of split links in 3D and 4D.
problem Homotopy type of embedding spaces of split links.
method Semi-simplicial space of separating systems, homotopy equivalence.
result Simple description of fundamental group of embedding space.
We show that any number of disjointly embedded 2-spheres in 4-space can be pulled apart by a link homotopy, ie, by a motion in which the 2-spheres stay disjoint but are allowed to self-intersect.
New TC variant dTC better fits motion planning for some systems.
problem Improving motion planning for autonomous systems.
method Defined and computed new homotopy invariant dTC.
result dTC and dcat provide better motion planning solutions.
Survey on manifold complexities and motion planning in robotics.
problem Understanding topological complexities of manifolds in robotic motion planning.
method Overview of topological complexities, geodesic motion planning, and connections to critical point theory.
result Estimation of motion planning complexity using Riemannian geometry and critical point theory.
The topological complexity TC(X) is a numerical homotopy invariant of a topological space X which is motivated by robotics and is similar in spirit to the classical Lusternik-Schnirelmann category of X. Given a mechanical system with configuration space X, the invariant TC(X) measures the complexity of all possible mot…
Study Brownian loops on hyperbolic surfaces, linking to Selberg zeta function.
problem Understanding Brownian loops on hyperbolic surfaces and their relation to Selberg zeta function.
method Computed mass of loops and related to Selberg zeta function for geometrically finite surfaces.
result Relate total loop mass to Selberg zeta function, providing probabilistic interpretations of determinants.
New metric invariant GC connects to TC, with applications in robotics.
problem Understanding motion planning in metric spaces.
method Introducing geodesic complexity (GC) as a new invariant.
result GC and topological complexity (TC) coincide in many cases but can be distinguished.
To each three-component link in the 3-sphere, we associate a geometrically natural characteristic map from the 3-torus to the 2-sphere, and show that the pairwise linking numbers and Milnor triple linking number that classify the link up to link homotopy correspond to the Pontryagin invariants that classify its charact…
Long-term human motion can be represented as a series of motion modes---motion sequences that capture short-term temporal dynamics---with transitions between them. We leverage this structure and present a novel Motion Transformation Variational Auto-Encoders (MT-VAE) for learning motion sequence generation. Our model j…
Introduces Motion Programs for better video analysis of human motion.
problem Current video analysis focuses on raw pixels or keypoints, missing higher-level motion primitives.
method Introduces Motion Programs as a neuro-symbolic representation of motions as a composition of high-level primitives.
result Motion Programs accurately describe diverse human motions and improve downstream tasks.
Study motion planning for points avoiding obstacles in a plane.
problem Avoiding collisions for multiple points in a plane with unknown obstacles.
method Algebraic and topological tools for motion planning.
result New topological complexity for planar motion planning.
Programmatic Motion Concepts learn human actions from paired videos.
problem Learning motion concepts from paired video and action sequences.
method Semi-supervised learning architecture for hierarchical motion representation.
result Outperforms established baselines, especially in small data settings.
Unified framework for human motion generation on Riemannian manifolds.
problem Learning valid human motion in Euclidean spaces.
method Riemannian Motion Generation (RMG) on product manifolds, Riemannian flow matching.
result Achieves state-of-the-art FID (0.043) on HumanML3D and surpasses strong baselines on MotionMillion.
Study on determinants of unitary Brownian motion and their asymptotic laws.
problem Understanding determinants of unitary Brownian motion and their behavior over time.
method Using Stiefel fibration and skew-product decomposition of the Stiefel Brownian motion.
result Prove asymptotic laws for determinants of block entries of unitary Brownian motion.
New framework predicts diverse, contextually plausible 3D human motions.
problem Predicting multiple plausible future 3D poses given observed poses.
method Developed a new variational framework that conditions latent variable on past observation to encourage relevant information.
result Our approach generates motions of higher quality and preserves contextual information.
Homotopy on nanophrases is an equivalence relation defined using some data called a homotopy data triple. We define a product on homotopy data triples. We show that any homotopy data triple can be factorized into a product of prime homotopy data triples and this factorization is unique up to isomorphism and order. If a…
New examples of manifolds that are homotopy but not simple homotopy equivalent.
problem Characterizing simple homotopy types of even dimensional manifolds.
method Using algebraic K-theory, surgery obstruction map, and homotopy automorphisms.
result Construction of infinite families of manifolds that are homotopy equivalent but not simple homotopy equivalent.
By considering homotopies that preserve the stratification, one obtains a natural notion of homotopy for stratified spaces. In this short note, we introduce invariants of stratified homotopy, the stratified homotopy groups. We show that they satisify a stratified version of Whitehead's theorem. As an example, we introd…
Data-driven modelling and synthesis of motion is an active research area with applications that include animation, games, and social robotics. This paper introduces a new class of probabilistic, generative, and controllable motion-data models based on normalising flows. Models of this kind can describe highly complex d…
Classifies colored links and spatial graphs up to colored link-homotopy.
problem Classifying colored links and spatial graphs up to colored link-homotopy.
method Using Habegger-Lin theory for colored string links, and extending to colored links and spatial graphs.
result Classification of colored links and spatial graphs up to colored link-homotopy.
Study refracted skew Brownian motion, find densities and asymptotics.
problem Modeling and analyzing refracted skew Brownian motion.
method Perturbation approach to find potential densities, transition density, and asymptotic behaviors.
result Expressions and asymptotic behaviors of refracted skew Brownian motion.
Neural network predicts vessel motions with high accuracy.
problem Real-time prediction of heave and surge motions for improved performance and safety.
method Developed an LSTM-based machine learning model trained on measured waves and motion data.
result The model predicts vessel motions up to 46.5 seconds into the future with an average accuracy of 90%.
Paper proves homotopy braid group properties over integers and three strands.
problem Understanding homotopy braid group properties.
method Proved linearity over integers and torsion freeness for three strands.
result Homotopy braid group on three strands is torsion free.
Study fractal dimension for motion without crossing a subset.
problem Fractal dimension of a subset X in R^n for motion without crossing.
method Analyzes fractal dimension of subset X in R^n.
result Determines conditions for motion without crossing a subset.
Let E E E be a closed set in the Riemann sphere C ^ \widehat{\mathbb{C}} C . We consider a holomorphic motion φ φ φ of E E E over a complex manifold M M M , that is, a holomorphic family of injections on E E E parametrized by M M M . It is known that if M M M is the unit disk Δ Δ Δ in the complex plane, then any holomorphic motion of E E E ove…
Neural network estimates rigid motion in stroke imaging to improve image quality.
problem Rigid patient motion during C-arm CBCT imaging reduces image quality.
method Neural network trained to regress reprojection error based on image information.
result Neural network outperforms entropy-based method in motion estimation.
Study cohomological equation for robotic screw motions on SE(3).
problem Understanding obstruction phenomena in robotic rigid-body motion.
method Combining Fourier analysis and Peter-Weyl theory, reduce to finite-dimensional linear transport systems.
result Explicit screw motion illustrates resonance conditions and finite-dimensional obstructions.
New examples of manifolds with similar homotopy but different simple homotopy types.
problem Characterizing groups for which high-dimensional manifolds can be homotopy equivalent but not simple homotopy equivalent.
method Construction of doubles of thickenings and use of a formula for Whitehead torsion.
result Examples of high-dimensional manifolds exist for any finitely presented group with a nontrivial Whitehead group involution.
New approach for obstacle avoidance in robotics using learned representations.
problem Challenges in sensor-based motion planning for new and dynamic environments.
method Proposes a new obstacle representation using PointNet architecture trained jointly with policies for obstacle avoidance.
result Significant improvements in accuracy and efficiency compared to state of the art.
V. Turaev introduced the theory of topology of words and phrases in 2005. This is a combinatorialy extension of the theory of virtual knots and links. In this paper we generalize the notion of homotopy of words and phrases and we give geometric meanings of the generalized homotopy of words. Moreover using the generaliz…
The study shows how stabilizing manifolds with projective spaces affects their homotopy structure.
problem Understanding the homotopy of manifolds stabilized by projective spaces.
method Trace the effect of surgery on product manifolds, showing a loop homotopy decomposition after localization.
result A loop homotopy decomposition of a manifold after stabilization by a projective space is provided.
Paper defines multi-dimensional fractional Brownian motion under volatility uncertainty.
problem Volatility uncertainty in fractional Brownian motion.
method Definition and study of multi-dimensional fractional Brownian motion (G-fBm) with Hurst index.
result First results on stochastic calculus for G-fBm with Hurst index > 0.5.
The paper presents a method to reduce arm motion complexity for prosthetics and robotics.
problem Reducing the complexity of human arm motions for robotic and prosthetic control.
method Data-driven techniques including DTW, DBA, Ward's distance, batch-DTW, and fPCA.
result Representative motion clusters and averages for different arm DOF levels.
New polynomials detect non-rotatable knotoid shapes.
problem Detecting non-rotatable knotoid shapes.
method Defined homotopy index polynomials for knotoids.
result Homotopy polynomials detect non-rotatable spherical knotoids.
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.
Researchers calculate the Laplace transform of a geometric Brownian motion integral.
problem Calculating the Laplace transform of a specific integral functional of geometric Brownian motion.
method Analytical calculation of the Laplace transform of the cumulative distribution and probability density functions.
result The Laplace transform of the integral functional of geometric Brownian motion is derived.
Researchers created a continuous Markov martingale that mimics Brownian motion but lacks the strong Markov property.
problem Constructing a continuous Markov martingale with Brownian marginals that misses the strong Markov property.
method Developed a new approach to create a continuous Markov martingale that differs from Brownian motion in terms of the strong Markov property.
result A continuous Markov martingale with Brownian marginals that lacks the strong Markov property was successfully constructed.
We consider n n n -dimensional discrete motions such that any two neighbouring positions correspond in a pure rotation ("rotating motions"). In the Study quadric model of Euclidean displacements these motions correspond to quadrilateral nets with edges contained in the Study quadric ("rotation nets"). The main focus of ou…
Improved vehicle motion prediction with uncertainty estimation.
problem Robust motion prediction for autonomous vehicles, especially under distributional shift.
method Presented an approach significantly improving the benchmark and taking 2nd place on the leaderboard.
result Significantly improved motion prediction and uncertainty measurement.
Edge-homotopy and vertex-homotopy are equivalence relations on spatial graphs which are generalizations of Milnor's link-homotopy. We introduce some edge (resp. vertex)-homotopy invariants of spatial graphs by applying the Sato-Levine invariant for the 2-component constituent algebraically split links and show examples…
The paper explores representations of graph manifolds to Seifert motion groups.
problem Existence of faithful representations of graph manifolds to Seifert motion groups.
method Discussion and proof of non-existence of certain representations.
result Graph manifolds can have virtually no faithful representations to the Seifert motion group.
Geodesic walks converge to Brownian motion on Finsler manifolds.
problem Understanding random walks on Finsler manifolds.
method Analyzing convergence of geodesic random walks to diffusion processes.
result The Brownian motion on a Riemannian metric is a key result.
A framework for computing holonomy groups of hybrid systems to achieve forward motion.
problem Achieving forward motion from periodic leg motion.
method Developing a framework for computing holonomy groups of hybrid systems.
result Computing holonomy groups of hybrid systems to achieve non-zero net motion.
Characterizes compact complex surfaces with finite homotopy rank-sum.
problem Compact complex surfaces with finite homotopy rank-sum.
method Characterization and proof of Steinness of universal cover.
result Smooth compact complex Kaehler surfaces with finite homotopy rank-sum.
Two approaches study the homotopy of blow ups in algebraic and symplectic geometry.
problem Investigate the homotopy of blow ups in algebraic and symplectic geometry.
method Develops fibrewise surgery theory and a purely homotopy theoretic approach.
result Obtained homotopy decompositions of the based loop space on blow ups.
Study shows equivariant Khovanov homotopy types are equivalent.
problem Understanding equivariant structures in Khovanov homotopy types.
method Investigates group actions on homotopy coherent diagrams to prove equivalence.
result Equivariant Khovanov homotopy types are equivariantly stably homotopy equivalent.
We explore homotopies in quantum field theory formalism.
problem Constructing homotopies in Batalin-Vilkovisky formalism.
method Review and construction of homotopies from renormalization group flow and gauge fixing changes.
result Constructing spans of quantum master actions with isomorphic effective actions using homotopies.