arXiv research

Study orders of canonical bundles over graph configuration spaces.

Classifies graph configuration spaces homeomorphic to manifolds.

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…

The study examines when a section exists for graph configuration spaces.

Proves conjecture on graph configuration spaces' complexity.

Paper addresses hidden faces in configuration space integrals for embeddings.

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…

Proves planar graphs' configuration spaces have highest topological complexity.

The paper calculates asymptotic Betti numbers and homology multiplicities for graph configuration spaces.

Defines a new invariant from graph configurations in three-manifolds.

Researchers analyze geodesic complexity in robot paths on tree graphs.

Optimizing the execution time of tensor program, e.g., a convolution, involves finding its optimal configuration. Searching the configuration space exhaustively is typically infeasible in practice. In line with recent research using TVM, we propose to learn a surrogate model to overcome this issue. The model is trained…

Researchers create a model for surface point configurations.

Study on planar graph braid groups' second homology.

New invariant counts graph configurations in 3D manifolds.

The paper calculates the Hilbert polynomials for configuration spaces over graphs with a short circumference.

Surveying topological complexity of graph configurations, unifying traditional and modern approaches.

Graph neural networks fail to distinguish certain 3D atom configurations.

Graph braid groups' complexity stabilizes for most graphs.

A new overlapping space solves the configuration search problem for graph embeddings.

For Gamma a finite, connected metric graph, we consider the space of configurations of n points in Gamma with a restraint parameter r dictating the minimum distance allowed between each pair of points. These restricted configuration spaces come up naturally in topological robotics. In this paper, we study the homotopy,…

Two graph homologies help compute embedding space.

The abstract formulates and proves a categorification of Robertson's conjecture.

Extends Heisenberg homology to ribbon graphs.

SpatialSim benchmarks machine learning in recognizing object spatial configurations.

Configuration spaces of distinct labeled points on the plane are of practical relevance in designing safe control schemes for Automated Guided Vehicles (robots) in industrial settings. In this announcement, we consider the problem of the construction and classification of configuration spaces for graphs. Topological da…

The paper classifies when certain graph braid groups are 3-manifold groups.

Paper rigorously defines Feynman graph integrals on Kähler manifolds.

Paper detects non-trivial cycles in embedding spaces using graph integrals.

Study lens spaces' definite fillings, classifying those with specific inequalities.

The study finds a subgroup of graph braid groups that is a direct product of non-abelian free groups.

Hass and Scott's example of a 4-valent graph on the 3-punctured sphere that cannot be realized by geodesics in any metric of negative curvature is generalized to impossible configurations filling surfaces of genus $n$ with $k$ punctures for any $n$ and $k$.

Signals are submanifolds; bounds on energy calculated.

Suppose that $C=(C_1,..., C_m)$ is a configuration of 2-dimensional symplectic submanifolds in a symplectic 4-manifold $(X,ω)$ with connected, negative definite intersection graph $Γ_C$. We show that by replacing an appropriate neighborhood of $\cup C_i$ with a smoothing $W_S$ of a normal surface singularity $(S, 0)$ w…

A linkage is a finite graph with lengths assigned to each edge. A planar realization is a map to the plane which preserves edge lengths. It can be thought of as a mechanical device formed from stiff rods and rotating joints. We look at the configuration space of all planar realizations of a linkage (following work of K…

The paper constructs non-trivial cocycles for long embeddings with more than one loop.

We study the problem of computing the homology of the configuration spaces of a finite cell complex $X$. We proceed by viewing $X$, together with its subdivisions, as a subdivisional space--a kind of diagram object in a category of cell complexes. After developing a version of Morse theory for subdivisional spaces, we …

New method generates equilibrium glass configurations efficiently.

Bott and Taubes used integrals over configuration spaces to produce finite-type a.k.a. Vassiliev knot invariants. Cattaneo, Cotta-Ramusino and Longoni then used these methods together with graph cohomology to construct "Vassiliev classes" in the real cohomology of spaces of knots in higher-dimensional Euclidean spaces,…

Holomorphic analogs of Feynman integrals are shown to be finite.

Let K be the space of long j-knots in R^n. In this paper we introduce a graph complex D and a linear map I from D to the de Rham complex of K via configuration space integral, and prove that (1) when both n>j>=3 are odd, the map I is a cochain map if restricted to graphs with at most one loop component, (2) when n-j>=2…

We propose a new method of computing cohomology groups of spaces of knots in $\R^n$, $n \ge 3$, based on the topology of configuration spaces and two-connected graphs, and calculate all such classes of order $\le 3.$ As a byproduct we define the higher indices, which invariants of knots in $\R^3$ define at arbitrary si…

Semi-supervised learning (SSL) is effectively used for numerous classification problems, thanks to its ability to make use of abundant unlabeled data. The main assumption of various SSL algorithms is that the nearby points on the data manifold are likely to share a label. Graph-based SSL constructs a graph from point-c…

These introductory lectures show how to define finite type invariants of links and 3-manifolds by counting graph configurations in 3-manifolds, following ideas of Witten and Kontsevich. The linking number is the simplest finite type invariant for 2-component links. It is defined in many equivalent ways in the first sec…

A venerable problem in combinatorics and geometry asks whether a given incidence relation may be realized by a configuration of points and lines. The classic version of this would ask for algebraic lines over some field or possibly real pseudolines: embedded circles (isotopic to $RP^1$) in the real projective plane. In…

Graph neural networks improve topology control of power grids.

In the present paper we introduce Mobius energy for the embedded graphs and formulate its main properties. This energy is invariant under the action of the group generated by all inversions in three-dimensional real space. We study critical configurations for the angles at vertices of degree less than five, and discuss…

We study configuration spaces of linkages whose underlying graph are polygons with diagonal constrains, or more general, partial two-trees. We show that (with an appropriate definition) the oriented area is a Bott-Morse function on the configuration space. Its critical points are described and Bott-Morse indices are co…