Proves conjecture on graph configuration spaces' complexity.
problem Topological complexity of graph configuration spaces.
method Lower bound derived from insights into aspherical spaces.
result Proves Farber's conjecture on stable topological complexity.
New calculations of topological complexity for symplectic CW-complexes.
problem Calculating topological complexity for symplectic CW-complexes.
method Using atoroidal cohomology classes and CW-complexes, proving topological complexity for symplectic spaces.
result Every atoroidally symplectic CW-complex of dimension 2n has topological complexity 4n.
The paper explores conditions for topological rigidity in quotients of the Davis complex.
problem Understanding when quotients of the Davis complex are topologically rigid.
method Analyzing quotients of the Davis complex of right-angled Coxeter groups and conditions on defining graphs.
result Introduction of infinitely many infinite topologically rigid subclasses.
The paper develops algorithms and topological invariants for distinguishing dynamic systems.
problem Distinguishing the topological type of surfaces and functions in dynamic systems.
method Construction of algorithms and topological invariants using discrete topological structures.
result The development of discrete topological structures for topological equivalence of dynamic systems.
The article applies Lusternik-Schnirelmann theory to establish lower bounds on critical points using sequential and parametrized topological complexity.
problem Establishing lower bounds on the number of critical points of functions using topological complexity.
method Applying Lusternik-Schnirelmann theory to sequential and parametrized topological complexity.
result Established various lower bounds on the number of critical points using sequential and parametrized topological complexity.
A new approach uses circuit topology to study complex polymer interactions.
problem Understanding structural phase transitions in entangled polymer systems.
method Braided circuit topology framework for multiple-chain systems.
result Circuit topological motif fractions are effective order parameters for structural transitions.
Constructs algorithms to recognize and classify 2D surfaces.
problem Recognizing and classifying 2D surfaces in dynamic systems.
method Discrete topological structures and algorithms for simplicial and CW-complexes.
result Determines the topological type of 2-manifolds.
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.
We present a new approach to equivariant version of the topological complexity, called a symmetric topological complexity. It seems that the presented approach is more adequate for the analysis of an impact of symmetry on the the motion planning algoritm than the one introduced and studied by Colman and Grant. We show …
CT improves neural network performance on cell complex data.
problem Improving predictive performance of neural networks on complex data.
method Introducing the Cellular Transformer (CT) that generalizes graph-based transformers to cell complexes.
result CT achieves state-of-the-art performance on cell complex datasets without complex enhancements.
We present some results supporting the Iwase-Sakai conjecture about coincidence of the topological complexity TC(X) and monoidal topological complexity TCM(X). Using these results we provide lower and upper bounds for the topological complexity of the wedge X∨Y. We use these bounds to give a counterexample t…
We provide an upper bound on the topological complexity of twisted products. We use it to give an estimate TC(X)≤TC(π1(X))+dimX of the topological complexity of a space in terms of its dimension and the complexity of its fundamental group.
Study topological invariants of complexes for Riemannian manifolds.
problem Understanding topological properties of Riemannian manifolds.
method Analyzing Betti numbers and Euler characteristic of Vietoris-Rips and Čech complexes.
result Betti curve converges to manifold's Betti number within a scale parameter interval.
Topological complexity for closed 1-forms
problem Topological complexity for closed 1-forms
method Introduce and study a corresponding version of topological complexity
result Establish analogues of basic properties of ordinary topological complexity
Lecture notes on curves in complex projective plane from a topological viewpoint.
problem Understanding curves in complex projective plane from a topological perspective.
method Topological analysis of curves in complex projective plane.
result Curves in complex projective plane have unique topological properties.
Study proves topological complexity and LS-category inequalities for specific groups and manifolds.
problem Proving inequalities for topological complexity and LS-category of specific groups and manifolds.
method Analyzing torsion free hyperbolic and nilpotent groups, lens spaces, using inequalities and counter-examples.
result Proves inequalities for topological complexity and LS-category of specific groups and manifolds.
We expand Topological Field Theory on some special CW-complexes (brane complexes). This Brane Topological Field Theory one-to-one corresponds to infinite dimensional Frobenius Algebras, graduated by CW-complexes of lesser dimension. We define general and regular Hurwitz numbers of brane complexes and prove that they ge…
Proves planar graphs' configuration spaces have highest topological complexity.
problem Proving Farber's conjecture for planar graphs.
method Generic maximality argument for topological complexities.
result Generic maximality of topological complexities for planar graphs.
In this paper we study a notion of topological complexity for the motion planning problem. The topological complexity is a number which measures discontinuity of the process of motion planning in the configuration space X. More precisely, it is the minimal number k such that there are k different motion planning rules,…
Graph braid groups' complexity stabilizes for most graphs.
problem Stabilization of topological complexity in graph braid groups.
method Geometric lower bounds on configuration spaces.
result Topological complexity stabilizes for most graphs.
Study numerical invariants under retraction maps between topological spaces.
problem Understand behavior of invariants like cohomological dimensions under retractions.
method Introduced a notion of retraction and studied several numerical invariants.
result Proved inequalities between invariants hold under retractions.
Studies amenable category's monotonicity and its relation to topological complexity.
problem Monotonicity of amenable category for degree-one maps.
method Uses amenable covers and compares with topological complexity.
result Establishes a relation between amenable category and topological complexity.
Survey on the topology of singular foliations in complex 2-space.
problem Understanding the topology of singular foliations in complex 2-space.
method Overview and survey of existing research.
result Overview of current knowledge on foliation singularities.
Study of universal complexes in toric topology with applications in category theory.
problem Properties and applications of universal complexes in toric topology.
method Combinatorial and topological analysis of X(Fpn) and K(Fpn). result Lusternick-Schnirelmann categories of moment angle complexes calculated for universal complexes.
The study finds conditions for nonmaximal topological complexity of manifolds with abelian fundamental groups.
problem Conditions for nonmaximal topological complexity of manifolds with abelian fundamental groups.
method Analyzes conditions and examples to generalize results on topological complexity and Lusternik-Schnirelmann category.
result Generalizes results on topological complexity and Lusternik-Schnirelmann category for manifolds with abelian fundamental groups.
New method estimates causal effects in complex spaces using topological structures.
problem Challenges in estimating causal effects in non-Euclidean spaces.
method Developed a topological causal inference framework using power-weighted silhouette functions of persistence diagrams.
result Successfully quantifies topological treatment effects across various complex outcomes.
New topological complexity measures for neural networks.
problem Measuring complexity of neural network functions.
method Generalized piecewise-linear Morse theory applied to ReLU networks.
result Local complexity can be arbitrarily high.
A new method for state estimation on complex networks.
problem Reconstructing latent dynamics from multivariate time-series on topological cell complexes.
method Topology-aware state space framework derived from stochastic partial differential equations, with state evolution following heat-like topological diffusion.
result The proposed method successfully recovers latent states and topological structures in real-world networks.
A new deep learning framework for topological data.
problem Developing models for data on complex topological domains.
method Introducing combinatorial complexes and developing attention-based CCNNs.
result CCNNs outperform existing models in tasks involving mesh shape analysis and graph learning.
Study the complexity of horizontality in 4-torus vector bundles.
problem Classify topological holonomy groups in SO(3).
method Analyze twistor spaces and oriented vector bundles over 2-torus.
result Discover many topological holonomy groups in SO(3) with noncommutative pairs.
The Lusternik-Schnirelmann category and topological complexity are important invariants of manifolds (and more generally, topological spaces). We study the behavior of these invariants under the operation of taking the connected sum of manifolds. We give a complete answer for the LS-categoryof orientable manifolds, $\c…
Locally flat submanifolds have finite CW complex complements.
problem Understanding the structure of manifold complements.
method Direct proof using homotopy equivalence and CW complexes.
result Complements of locally flat submanifolds are finite CW complexes.
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 paper classifies Poincaré complexes as topological manifolds.
problem Classifying Poincaré complexes as topological manifolds.
method Using spherical fibrations and CW-complexes, the paper proves stability and homotopy equivalence.
result A sufficient condition for Poincaré complexes to be homotopy types of topological manifolds.
We show that the topological complexity of a finitely generated torsion free hyperbolic group π with $\cdπ=n$ equals 2n.
Lower bound for complexity of finding flex points on cubic curves.
problem Finding flex points on cubic plane curves.
method Bounding the Schwarz genus of a cover associated to the problem.
result Lower bound for topological complexity close to optimal.
Paper calculates topological complexity of robot movement in narrow aisles.
problem Determining minimum number of scenarios for robot movement in a narrow strip.
method Examined cohomology ring of ordered configuration space to find lower bound.
result Lower bound for minimum number of cases in robot movement program.
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…
Novel algorithm learns sparse signal representations over topological spaces.
problem Sparse representation of signals over combinatorial topological spaces.
method Leveraging Hodge theory, the paper embeds topology into a dictionary structure via concatenated sub-dictionaries, each as a polynomial of Hodge Laplacians, and optimizes the dictionary coefficients and sparse signal representation via iterative alternating algorithms.
result Efficiently learned sparse representations and underlying relational structure of topological signals.
We combine Freedman's topology with Eliashberg's holomorphic theory to construct Stein neighborhood systems in complex surfaces, and use these to study various notions of convexity and concavity. Every tame, topologically embedded 2-complex K in a complex surface, after C^0-small topological ambient isotopy, is the int…
Proposes a constraint for deep clustering to handle both simple and complex topologies.
problem Limited prior knowledge for deep clustering methods to perform well on complex topologies.
method Introduces a constraint using symmetric InfoNCE to enhance deep clustering performance.
result The constraint improves deep clustering methods' performance on both simple and complex topologies.
We prove that the topological complexity of (a motion planning algorithm on) the complement of generic complex essential hyperplane arrangement of n hyperplanes in an r-dimensional linear space is min{n+1,2r}.
Proposes a probabilistic framework for stationary topological signals on simplicial complexes.
problem Complex data structures require new models and tools.
method Generalizes stationarity to topological signals on simplicial complexes.
result Defines topological power spectral density (PSD) for stationary signals.
Researchers found the global topology of the Eisenstein-Picard modular surface.
problem Understanding the global topology of the Eisenstein-Picard modular surface.
method Quotient space of the complex hyperbolic plane by the modular group.
result Determined the global topology of the Eisenstein-Picard modular surface as a 4-orbifold.
Study topological G₂ and Spin(7) strings at 1-loop using double complexes.
problem Calculate topological string partition functions at 1-loop.
method Define double complexes for supersymmetric backgrounds using generalised geometry, compute partition functions as alternating products of determinants of Laplacians.
result Reproduce known results for G₂ string and predict for Spin(7) string.
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.
The study examines the topology of complements of polytopal skeletons.
problem Characterizing topological properties of polytopal complexes and their skeletons.
method Constructing a long exact sequence relating homologies of skeleton complements and links of faces.
result Characterizations of Cohen-Macaulay and Leray complexes, stacked balls, and neighbourly spheres in terms of skeleton complements.
We introduce a variant of Farber's topological complexity, defined for smooth compact orientable Riemannian manifolds, which takes into account only motion planners with the lowest possible "average length" of the output paths. We prove that it never differs from topological complexity by more than 1, thus showing th…