We study configurations of immersed curves in surfaces and surfaces in 3-manifolds. Among other results, we show that primitive curves have only finitely many configurations which minimize the number of double points. We give examples of minimal configurations not realized by geodesics in any hyperbolic metric.
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48 results for “configuration results”
Study shows configuration spaces' homological dimension increases monotonically.
problem Understanding the homological properties of configuration spaces of manifolds.
method Analyzing the homological monotonicity of unordered configuration spaces of manifolds.
result Homological dimension of configuration spaces increases monotonically in each degree.
We consider the problem of configuring general-purpose solvers to run efficiently on problem instances drawn from an unknown distribution. The goal of the configurator is to find a configuration that runs fast on average on most instances, and do so with the least amount of total work. It can run a chosen solver on a r…
Tripod configurations of plane curves, formed by certain triples of normal lines coinciding at a point, were introduced by Tabachnikov, who showed that closed convex curves possess at least two tripod configurations. Later, Kao and Wang established the existence of tripod configurations for closed locally c…
Solves Plateau-Douglas problem for singular configurations in general metric spaces.
problem Existence of minimal surfaces for singular configurations.
method Generalized approach via minimal sequences in metric spaces.
result Existence of minimal surfaces for singular configurations in general metric spaces.
Decouples homotopy quotients of generalised configuration spaces on surfaces.
problem Homological stability of generalised configuration spaces on surfaces.
method Analyzes actions of diffeomorphism groups and uses homotopy quotients.
result Decouples theorem for homology of homotopy quotients on surfaces.
We prove a conjecture of Crapo and Penne which characterizes isotopy classes of skew configurations with spindle-structure. We use this result in order to define an invariant, spindle-genus, for spindle-configurations. We also slightly simplify the exposition of some known invariants for configurations of skew lines an…
We prove homological stability for sequences of "oriented configuration spaces" as the number of points in the configuration goes to infinity. These are spaces of configurations of n points in a connected manifold M of dimension at least 2 which 'admits a boundary', with labels in a path-connected space X, and with an …
This paper extends homological stability results for configuration spaces of manifolds.
problem Homological stability of configuration spaces of manifolds.
method Analyzing the cohomology of configuration spaces of manifolds, focusing on stability in odd and even degrees.
result The stable range for homology groups of configuration spaces depends on the dimension of the manifold and the number of configuration points.
Researchers calculate cohomological dimensions of manifold configuration spaces, proving arithmeticity and providing bounds.
problem Calculating the cohomological dimensions of configuration spaces of manifolds.
method Defined a reduced Chevalley Eilenberg complex and provided precise formulas and bounds.
result Arithmeticity of cohomological dimensions in configuration spaces of manifolds with non-trivial co-dimension one cohomology groups.
Using results on the topology of moduli space of polygons [Jaggi, 92; Kapovich and Millson, 94], it can be shown that for a planar robot arm with segments there are some values of the base-length, , at which the configuration space of the constrained arm (arm with its end effector fixed) has two disconnected com…
In this paper we determine the topological complexity of configuration spaces of graphs which are not necessarily trees, which is a crucial assumption in previous results. We do this for two very different classes of graphs: fully articulated graphs and banana graphs. We also complete the computation in the case of tre…
Data-driven configuration tuning of glmnet for balancing accuracy and computational efficiencystat.CO
Optimizes glmnet configuration for better accuracy and efficiency.
problem Inappropriate glmnet configuration leads to inaccurate solutions and increased computation time.
method Data-driven framework using neural networks to predict accuracy and computation time from dataset characteristics and configuration.
result Automatic selection of optimal configuration maximizing accuracy under a time constraint.
We use the compression theorem (arxiv:math.GT/9712235) cf section 7, to prove results for equivariant configuration spaces analogous to the well-known non-equivariant results of May, Milgram and Segal.
Computes fundamental groups of restricted configuration spaces.
problem Understanding the structure of configuration spaces after removing hypersurfaces.
method Fibration over unordered configuration spaces of points, computation of fundamental groups.
result Fundamental groups of restricted configuration spaces computed in small dimensions.
We complete the classification of supersymmetric configurations of two M5-branes, started by Ohta and Townsend. The novel configurations not considered before are those in which the two branes are moving relative to one another. These configurations are obtained by starting with two coincident branes and Lorentz-transf…
Summary of pure cactus groups and circle points.
problem Understanding pure cactus groups and configuration spaces.
method Summarizes previous work on the topic.
result Summary of results from previous papers and thesis.
New proof of a 111-year-old result using gauge theory.
problem Proving configurations in the list of unavoidable configurations are reducible.
method Filtered 3- and 4-color homology, gauge theory, state-reducibility.
result Birkhoff diamond is reducible using gauge theory.
Algorithm calculates stable multiplicities in cohomology of configuration spaces.
problem Computing stable multiplicities of irreducible representations in cohomology.
method Developed an algorithm to compute stable multiplicities of families of irreducible representations.
result Computed stable multiplicities for all Young diagrams with 23 boxes up to degree 50.
We study the configuration space of equilateral and equiangular spatial hexagons for any bond angle by giving explicit expressions of all the possible shapes. We show that the chair configuration is isolated, whereas the boat configuration allows one-dimensional deformations which form a circle in the configuration spa…
Given a polarized complex manifold, projection of a torus-equivariant test configuration to holomorphic vector fields was introduced by G. Székelyhidi, as the limit of the associated -actions. We show that there actually holds the moment convergence of the weight distributions. Our analytic approach at th…
Exact universal interpolation property for landmark configurations in Euclidean space.
problem Representing and deforming landmark configurations through flows of vector fields.
method Explicitly describe vector fields for exact universal interpolation property in all dimensions.
result Achieve controllability by combining constant and polynomial vector fields.
Non-trivial action on surface configuration homology.
problem Understanding the Johnson filtration's action on surface configuration homology.
method Action of mapping class group on configuration space homology, focusing on Johnson filtration stages.
result Non-trivial action on the stage of the Johnson filtration for all and .
The paper classifies holomorphic maps between Riemann surface configuration spaces.
problem Classifying holomorphic maps between configuration spaces of Riemann surfaces.
method Group-theoretic rigidity results promoted to the space level.
result Complete classifications of holomorphic maps between configuration spaces of Riemann surfaces.
Introduces non-Archimedean metrics for pseudoeffective classes on Kähler manifolds.
problem Characterizing and approximating non-Archimedean metrics on pseudoeffective classes.
method Extending Ross-Witt Nyström correspondence to relative case, introducing flag configurations.
result Non-Archimedean finite energy metrics are approximable by flag configurations, and very general Ding energies are continuous.
Researchers create a model for surface point configurations.
problem No existing models for surface configuration spaces.
method Constructed a discrete combinatorial model for an oriented surface.
result Homotopy equivalence between configuration space and subcomplex of cube complex.
This article is the continuation of our first article (math/9901028). It shows how the zero-anomaly result of Yang implies the equality between the configuration space integral and the Kontsevich integral.
Machine learning algorithms are very sensitive to the hyperparameters, and their evaluations are generally expensive. Users desperately need intelligent methods to quickly optimize hyperparameter settings according to known evaluation information, and thus reduce computational cost and promote optimization efficiency. …
We study the Orchard relation for generic configurations of points in the plane (also called order types). We introduce infinitesimally-close points and analyse the relation of this notion with the Orchard relation. The second part of the paper deals with monochromatic configurations (for the Orchard relation). We give…
Quantum groups created from disk configuration space homologies.
problem Creating quantum groups from algebraic structures.
method Reconstructing quantum groups from homologies of configuration spaces of disks.
result New combinatorics and actual submanifolds of configuration spaces.
The paper improves bounds on how many squares can fit in a rectangle and still have stable homology.
problem Homological stability in the space direction of square configurations.
method Analyzing the ordered configuration space of squares in a rectangle.
result Most rectangles can be almost entirely filled with squares and still have stable homology.
A machine learning configuration refers to a combination of preprocessor, learner, and hyperparameters. Given a set of configurations and a large dataset randomly split into training and testing set, we study how to efficiently select the best configuration with approximately the highest testing accuracy when trained f…
Study orders of canonical bundles over graph configuration spaces.
problem Determining bundle orders for planar and nonplanar graphs.
method Analyzing configuration spaces of graphs to find bundle orders.
result Bundle orders are 2 for planar and 4 for nonplanar graphs.
The performance of modern machine learning methods highly depends on their hyperparameter configurations. One simple way of selecting a configuration is to use default settings, often proposed along with the publication and implementation of a new algorithm. Those default values are usually chosen in an ad-hoc manner t…
Detecting exotic spheres involves analyzing framed configuration spaces.
problem Detecting exotic spheres through homotopy type of truncated Disc-presheaves.
method Using gluing results for Disc-presheaves, Atiyah duality, and computations of mapping class groups.
result Conditions for detecting exotic spheres through framed configuration spaces.
Ensuring that all supposedly valid configurations of a software product line (SPL) lead to well-formed and acceptable products is challenging since it is most of the time impractical to enumerate and test all individual products of an SPL. Machine learning classifiers have been recently used to predict the acceptabilit…
Study on disk configurations in strips shows stability patterns.
problem Understanding stability patterns in disk configurations in strips.
method Finite presentation of rational homology groups, representation stability.
result Disk configuration space exhibits first-order representation stability.
Study maps surface configurations to Heisenberg homologies for mapping class groups.
problem Understanding Mapping Class Groups of punctured surfaces.
method Action of mapping classes on Heisenberg homologies of surface configurations.
result Representations of Mapping Class Groups derived from Heisenberg homologies.
We use rational formality of configuration spaces and the bar construction to study the cohomology of the space of braids in dimension four or greater. We provide a diagram complex for braids and a quasi-isomorphism to the de Rham cochains on the space of braids. The quasi-isomorphism is given by a configuration space …
Sharp upper bound for quasi polynomial degree of manifold configuration spaces.
problem Determining the exact degree of quasi-polynomial homology groups of configuration spaces.
method Analyzing extremal homology groups of unordered configuration spaces of manifolds.
result The upper bound for the degree of quasi-polynomials is sharp for every manifold.
We prove similar theorems concerning the structure of bundles involving complements of fiber-type hyperplane arrangements and orbit configuration spaces. These results facilitate analysis of the fundamental groups of these spaces, which may be viewed as generalizations of the Artin pure braid group. In particular, we r…
The square-peg problem is solved using configuration spaces and multijet transversality.
problem Proving that every simple closed curve in the plane has an odd number of inscribed squares.
method Using the multijet transversality theorem and configuration spaces, we find a dense set of smooth embeddings for which the configuration space of points is transverse to any submanifold.
result A dense family of smoothly embedded circles in the plane and in have an odd number of inscribed square-like quadrilaterals.
We show that every negative definite configuration of symplectic surfaces in a symplectic 4--manifold has a strongly symplectically convex neighborhood. We use this to show that, if a negative definite configuration satisfies an additional negativity condition at each surface in the configuration, and if the complex si…
The configuration space of the reduced Hamiltonian formulation of quantum gravity has been shown, for non-Ricci flat metrics, to be a higher-dimensional analogue of the Teichmüller space of conformal structures on a Riemann surface. In this article we show that the configuration space of conformal connection-dynamics i…
We compute the fundamental group of various spaces of Desargues configurations in complex projective spaces: planar and non-planar configurations, with a fixed center and also with an arbitrary center.
This expository article describes applications of topological configuration spaces to the control of robotic systems. In particular, we review recent work by the authors on configuration spaces of graphs. These are lovely spaces: we show for example that the configuration space of two points on the complete graph of fi…
We generalize Fulton and MacPherson's configuration space construction to weighted filtered manifolds.
problem Infinitesimal collision data in filtered manifolds with higher-order compatibility.
method Generalizing Fulton and MacPherson's blow-up approach to weighted arrangements of submanifolds.
result Smoothness of the weighted blow-up under reasonable assumptions.
A new overlapping space solves the configuration search problem for graph embeddings.
problem Configuring product spaces for graph embeddings is resource-intensive and impractical.
method Introducing overlapping spaces that share subsets of coordinates between different types of spaces (Euclidean, hyperbolic, spherical).
result Overlapping spaces achieve nearly optimal results without configuration tuning, reducing training time.