Survey on Kodaira dimension in 2D and 3D topology.
problem Various notions of Kodaira dimension in low dimensional topology.
method Survey and review of existing literature.
result Progress in understanding Kodaira dimension in low dimensions.
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169,341 papers · 148 categories
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48 results for “low dimensional topology”
This article sketches various ideas in contact geometry that have become useful in low-dimensional topology. Specifically we (1) outline the proof of Eliashberg and Thurston's results concerning perturbations of foliatoins into contact structures, (2) discuss Eliashberg and Weinstein's symplectic handle attachments, an…
Unified framework for 3D and 4D manifold and knot theory.
problem Unified understanding of manifold and knot theory.
method Unified correspondence between different subfields of low-dimensional topology.
result Unified algebraic manifestations of 3D and 4D manifold and knot theory.
Explains how knots relate to 4D shapes.
problem Understanding 4D shapes through knot theory.
method Combines knot theory with 4D manifold topology.
result Connects 4D shapes to knot theory and other geometries.
In connection with his interest in selfdistributive algebra, Richard Laver established two deep results with potential applications in low-dimensional topology, namely the existence of what is now known as the Laver tables and the well-foundedness of the standard ordering of positive braids. Here we present these resul…
Efficient method for computing twisted Alexander polynomials of Montesinos links.
problem Computing twisted Alexander polynomials for Montesinos links efficiently.
method Developed an efficient method to compute the twisted Alexander polynomial for Montesinos links using any linear representation.
result Formulas for multi-variable Alexander polynomials can be easily derived.
A neural network visualizes data structure and concepts.
problem Data visualization and concept understanding.
method Mixing autoencoder and classifier for multi-perspective visualization.
result The network produces different topological maps based on training as autoencoder or classifier.
Classical topological concepts are applied to understand high performance computing simulations of molecules writhing in three dimensional space. These simulations produce peta-bytes of floating point data, to describe 3 dimensional changes in molecular structure. A zero-th order analysis is achieved by viewing a compu…
Low-dimensional GKM theorymath.AT
Survey on GKM theory in low dimensions, highlighting combinatorics-geometry interplay.
problem Generalizing classical ideas from quasi-toric manifolds to torus actions.
method GKM theory applied to low-dimensional cases.
result Particularly fruitful interaction between geometry and combinatorics in low dimensions.
Notes on Khovanov and knot Floer theories' stable homotopy types.
problem Understanding stable homotopy types in Khovanov and knot Floer theories.
method Introduction to Khovanov and knot Floer theories' stable homotopy types.
result Introduction of stable homotopy types in Khovanov and knot Floer theories.
We review localization techniques for functional integrals which have recently been used to perform calculations in and gain insight into the structure of certain topological field theories and low-dimensional gauge theories. These are the functional integral counterparts of the Mathai-Quillen formalism, the Duistermaa…
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.
Abstracts discuss a common framework for constructing homology theories.
problem Developing homology theories in low-dimensional topology and geometry.
method Uses a common framework since the late 1980s to construct homology theories.
result Indicates the specific nature of the situation dictates the algebraic nature of chain groups.
Study knot singularities in Bogomolny equation solutions.
problem Understanding solutions with knot singularities.
method Analyzes the moduli space of solutions on R^3 with specific asymptotic conditions.
result Potential applications in low-dimensional topology and knot theory.
Survey on categorifying Jones polynomial.
problem Categorification of Jones polynomial.
method Not explicitly stated, likely involves algebraic and geometric categorification techniques.
result Significance and ramifications in geometry, algebra, and topology.
Mathematical treatment of sigma model's low energy theory.
problem Low energy effective theory of sigma model and its topology.
method Relates beta-function to Ricci curvature of target manifold.
result Recovery of Friedan's physical result.
Cobordism of wordsmath.CO
We introduce an equivalence relation, called cobordism, for words and study cobordism invariants of words inspired by methods of low-dimensional topology.
FibeRed reduces complex data dimensions while preserving topology.
problem Hard embedding of topologically complex datasets in low-dimensional Euclidean space.
method Modeling datasets with vector bundles, reducing fibers while preserving topology.
result FibeRed learns topologically faithful embeddings in lower dimensions than existing methods.
The paper studies bifurcations in discrete dynamical systems on manifolds.
problem Understanding bifurcations in discrete dynamical systems on manifolds.
method Topological techniques based on concentricity of manifolds.
result General result for attractors in n-dimensional manifolds.
Lecture notes on Heegaard Floer homology for beginners.
problem Introducing Heegaard Floer homology to newcomers.
method Illustrates algebraic structures via grid homology, then defines Heegaard Floer homology geometrically.
result Defines and describes key properties of Heegaard Floer homology.
L. Kauffman conjectured that a particular solution of the Chinese Rings puzzle is the simplest possible. We prove his conjecture by using low-dimensional topology and group theory. We notice also a surprising connection between the Chinese Rings and Habiro moves (related to Vassiliev invariants).
The purpose of this thesis is to study classical combinatorial objects, such as polytopes, polytopal complexes, and subspace arrangements, using tools that have been developed in combinatorial topology, especially those tools developed in connection with (discrete) differential geometry, geometric group theory and low-…
We simplify neural networks to 3D to study their topological changes.
problem Understanding how neural network layers affect low-dimensional topological invariants.
method Limiting each layer to a width of 3D space, tracking changes in linking numbers.
result ResNets and transformers are equally powerful in changing linking numbers.
A new metric captures shape information in manifold learning.
problem Capturing shape information in high-dimensional data.
method Metric based on angular changes along geodesic lines.
result Feasibility and merits of proposed dimensionality reduction scheme.
In low dimensional topology, we have some invariants defined by using solutions of some nonlinear elliptic operators. The invariants could be understood as Euler class or degree in the ordinary cohomology, in infinite dimensional setting. Instead of looking at the solutions, if we can regard some kind of homotopy class…
New algorithm improves topological stability in non-linear dimensionality reduction.
problem Topological instability in choosing nearest neighbors in Isomap.
method Uses point and its two nearest neighbors to find subspace and orthogonal complement, then adds new points based on distance and angle.
result Improves topological stability and reduces short-circuit errors.
This is an expository paper discussing various versions of Khovanov homology theories, interrelations between them, their properties, and their applications to other areas of knot theory and low-dimensional topology.
New algorithm constrains SOMs to create supervised low-dimensional mappings.
problem Creating supervised mappings in neural networks with known internal topology.
method Developed Supervised Topological Maps (STMs) by modifying SOMs to incorporate target distances.
result STMs allow for supervised generation of new data with known internal structure.
PERCEPT detects changes in high-dimensional data streams using topological data analysis.
problem Detecting changes in high-dimensional data streams, especially when embedded in a low-dimensional space.
method Leverages topological data analysis to learn embedded topology as a point cloud via persistence diagrams, then applies non-parametric monitoring for detecting changes.
result Demonstrates efficient detection of online changes from high-dimensional data streams.
Lecture notes on Lie groups and Chern-Simons theory for grad students.
problem Understanding Chern-Simons theory and its applications.
method Explains Lie groups and their relation to Chern-Simons theory.
result Motivated new fields in knot theory and topology.
Paper introduces a complete metric topology for low energy spaces.
problem Defining a topology for low energy spaces with prescribed singularity.
method Introduces a completely metrizable topology stronger than capacity convergence.
result Low energy spaces have a natural completely metrizable topology.
The results of the paper concern the topological structure of complete riemannian manifolds with cyclic holonomy groups and low-dimensional orientable complete flat manifolds. We also discuss related results such as the affine classification of orientable complete flat 4-manifolds, an algebraic criterion of an affine e…
Gordon-Litherland pairing connects combinatorics and topology.
problem Unifying quadratic forms in link theory.
method Picture proof using Kirby diagrams.
result Their theorem has numerous applications in low-dimensional topology.
We introduce a new class of possibly noncompact n-dimensional manifolds without boundary associated to finite data which we call topological automata. This class is large enough to contain many interesting examples of open 2-dimensional and 3-dimensional manifolds of interest to low-dimensional topologists. Our main re…
The paper tackles VAEs with manifold-valued latent variables.
problem VAEs struggle with non-trivial manifold topologies.
method Extended reparameterization trick to Lie groups, focusing on .
result Manifold-valued latent variables preserve topological structure.
Lagrangian Non-Intersectionsmath.SG
In this paper we use Floer theory to study topological restrictions on Lagrangian embeddings in closed symplectic manifolds. One of the phenomena arising from our results is ``homological rigidity'' of Lagrangian submanifolds. Namely, in certain symplectic manifolds, conditions on low dimensional topological invariants…
Study the periods mapping from hyperelliptic curves, revealing fiber topology.
problem Global topology of 2D fibers of the periods mapping.
method Decomposition of moduli space into polyhedra labeled by planar graphs.
result Investigation of low dimensional fibers of the periods mapping.
Classifies fake surfaces up to complexity 5.
problem Classifying fake surfaces for low-dimensional topology.
method Derived properties of fake surfaces, classified up to complexity 5.
result Proved conjectures about fake surfaces up to complexity 5.
New mathematical tools for studying knots and links.
problem Understanding knot and link diagrams using topological invariants.
method Introducing Khovanov Laplacian and Khovanov Dirac to study diagrams.
result The harmonic spectrum retains Khovanov homology invariants, while non-harmonic spectra reveal additional information.
Crossed homomorphisms and low dimensional representations of mapping class groups of surfacesmath.GT
Study of low dimensional representations of mapping class groups of surfaces, focusing on genus ≥ 7.
problem Classifying -dimensional complex linear representations of mapping class groups.
method Using twisted 1-cohomology groups and Morita's computation, a complete classification is given for .
result No irreducible linear representations of dimension for .
Turaev transformed knot theory and 3-manifold invariants.
problem Understanding knots and links invariants of 3-manifolds.
method Classical topology techniques and quantum topology.
result Introduced new ideas and tools in knot theory and 3-manifold invariants.
This paper tackles the topology of infinite-dimensional fiber spaces.
problem Classifying and understanding the topological structure of infinite-dimensional fiber spaces.
method Algebraic and differential topology, focusing on the homotopy type of the moduli space of fibrations.
result Explicit homotopy calculations for low-dimensional cases, completing the solution for dimensions up to three.
Graph Neural Networks solve topology problems in simple 3D models.
problem Deciding homeomorphism of 3-manifolds described by plumbing graphs.
method Supervised and reinforcement learning with Graph Neural Networks.
result High accuracy in determining homeomorphic 3-manifolds.
Proposes a model for identifying edges in low-rank dynamical networks.
problem Inability of conventional methods to handle low-rank dynamical networks.
method Low rank dynamical network model with causal Wiener filtering.
result Consistent method for estimating all network edges.
Study integral bounds for submanifolds in low codimension with topological implications.
problem Integral curvature bounds and topological obstructions for submanifolds.
method Integral curvature bounds in terms of Betti numbers, -pinched immersions, and pinched second fundamental form.
result Obtained topological obstructions for -pinched immersions and intrinsic obstructions for minimal submanifolds in spheres.
Survey of homology cobordism group with open problems.
problem Understanding the structure and behavior of homology cobordism groups.
method Review and discussion of recent results and open problems.
result Discussion of open problems in homology cobordism groups.
Study perturbs Dirac operators in any dimension, focusing on Majorana fermions.
problem Understanding perturbations of Dirac operators in various dimensions.
method Analyzes canonical perturbations of Dirac operators on Hermitian Clifford modules.
result Characterizes the low-energy spectrum of these operators on complete surfaces.
Chart autoencoders learn latent features preserving manifold topology and geometry, with robust denoising capabilities.
problem Learning low-dimensional latent features of high-dimensional data sampled near a manifold.
method Chart autoencoders encode data into latent features on charts, preserving manifold topology and geometry.
result Chart autoencoders achieve a squared generalization error of under proper network architectures.