This paper shows connections between two complex mathematical theories are equivalent.
problem Establishing equivalence between two complex mathematical theories.
method Using geometric quantisation and conformal field theory, the paper establishes equivalence between the Hitchin connection and the Knizhnik-Zamolodchikov connection.
result The Hitchin and Knizhnik-Zamolodchikov connections are projectively equivalent in genus zero.
Establishes equivalent formulations of connections in tangent categories.
problem Generalizing connections in tangent categories.
method Defined and generalized the notion of connection on differential bundles in tangent categories, provided equivalent formulations, and showed equivalences with specific morphisms and diagrams.
result Equivalent formulations of connections in tangent categories reduce the amount of specified structure and axioms required.
Affine equivalence of half-translation surfaces via saddle connection graphs.
problem Understanding affine equivalence of half-translation surfaces.
method Association of saddle connection graphs and investigation of their automorphism groups.
result Every isomorphism between saddle connection graphs is induced by an affine homeomorphism between the underlying half-translation surfaces.
We solve the local equivalence problem for sub-Riemannian structures on (2n + 1)-dimensional manifolds. We show that two sub-Riemannian structures are locally equivalent if and only if? their corresponding canonical linear connections are equivalent. When n = 1, these connections coincide with the generalized Tanaka-We…
We prove that an open 3-manifold proper homotopy equivalent to a geometrically simply connected polyhedron is simply connected at infinity, generalizing a theorem of V.Poenaru.
Constructs Lepage equivalents for arbitrary-order Lagrangians.
problem Creating Lepage equivalents for complex Lagrangians.
method Uses variational bicomplex and symmetric linear connections to construct Lepage equivalents satisfying the closure property.
result Shows how to extend global Lepage equivalents to ones satisfying the closure property.
Study Yang-Mills connections on surfaces via Fréchet reduction.
problem Characterize Yang-Mills connections on orientable closed surfaces.
method Fréchet reduction to define and constrain gauge equivalence classes of connections.
result Deduced stratified symplectic structure on unbased Yang-Mills connections.
The paper finds manifold structures on complex spaces.
problem Constructing manifold structures on highly connected Poincaré complexes.
method Constructing examples and determining homotopy types.
result Examples of highly connected Poincaré complexes are found to be homotopy equivalent to manifolds but not smooth.
Using the same method we provide negative answers to the following questions: Is it possible to find real equations for complex polynomials in two variables up to topological equivalence (Lee Rudolph) ? Can two topologically equivalent polynomials be connected by a continuous family of topologically equivalent polynomi…
Equivalence of second order differential operators in vector bundles studied.
problem Equivalence problem for second order linear differential operators in vector bundles.
method Description of rational invariants of symbols, finding connections associated with differential operators.
result Solving problems of local and global equivalency of differential operators.
This paper explains spectral clustering and its equivalence to PCA, breaking it into fully connected and multi-connected cases.
problem Understanding the mathematics behind spectral clustering and its equivalence to PCA.
method Dividing spectral clustering into two categories based on graph connectivity and proving the equivalence to PCA.
result Spectral clustering and PCA are equivalent, with specific proofs for fully connected and multi-connected graphs.
Models for self-equivalences and diffeomorphisms of manifolds.
problem Classifying spaces of self-equivalences and diffeomorphisms of manifolds.
method Construct rational models using equivariant algebraic methods.
result Formula for rational cohomology of classifying spaces.
Paper defines conditions for locally metric connections in plane bundles.
problem Conditions for locally metric connections in plane bundles.
method Necessary and sufficient conditions derived in local charts and globally in terms of Euler class.
result Global necessary condition for metric equivalence of connections.
We introduce the equation of n-dimensional totally geodesic submanifolds of a manifold E as a submanifold of the second order jet space of n-dimensional submanifolds of E. Next we study the geometry of n-Grassmannian equivalent connections, that is linear connections without torsion admitting the same equation of n-dim…
Paper shows equivalences between Dixmier-Douady bundles and differential 3-cocycles.
problem Understanding equivalences between Dixmier-Douady bundles and differential 3-cocycles.
method Focuses on the model of Dixmier-Douady bundles and provides equivalences between 2-stacks.
result Equivalence between Dixmier-Douady bundles and differential 3-cocycles of height 1.
Study logarithmic flat connections on principal bundles using Lie groupoids.
problem Classify flat connections on principal bundles with logarithmic singularities.
method Use tools from Lie groupoid theory to classify representations and establish van Kampen theorems.
result Obtain a functorial Riemann-Hilbert correspondence for logarithmic connections.
Saddle connection complexes are rigid under affine equivalence.
problem Characterizing the rigidity of saddle connection complexes.
method Proving simplicial isomorphisms between saddle connection complexes are induced by affine diffeomorphisms.
result Saddle connection complexes are complete invariants of affine equivalence classes of half-translation surfaces.
The study classifies affine surface connections and their moduli spaces.
problem Classifying affine surface connections and their moduli spaces.
method Analyzing the quasi-Einstein equation and using affine equivalence.
result Affine surface connections are classified up to linear and affine equivalence.
We show that the category of abelian gerbes over a smooth manifold is equivalent to a certain category of principal bundles over the free loop space. These principal bundles are equipped with fusion products and are equivariant with respect to thin homotopies between loops. The equivalence is established by a functor c…
The paper shows connections can be uniquely determined by their boundary data.
problem Determining unique connections from boundary measurements.
method Defined a Dirichlet-to-Neumann map for twisted Dirac Laplacians and showed its pseudodifferential properties.
result Equal Dirichlet-to-Neumann maps imply locally gauge equivalent connections.
Newly proves equivariant bundles equivalence to differential quotient stacks.
problem Equivalence of equivariant bundles and differential quotient stacks.
method Proved equivalence through Lie group G action on manifold M.
result Category equivalence of equivariant bundles and differential quotient stacks.
Every infinitely edge-connected graph has a minor of Farey graph or Tℵ0∗t.
problem Characterizing edge-connected graphs with specific minor properties.
method Analyzing the minor structure of infinitely edge-connected graphs.
result Infinitely edge-connected graphs contain Farey graph or Tℵ0∗t as a minor. Maps connect K-theory and L-theory for complex C*-algebras.
problem Establishing a connection between K-theory and L-theory for C*-algebras.
method Proving a natural map of spectra between connective topological K-theory and connective algebraic L-theory.
result A natural equivalence between periodic K- and L-theory spectra after inverting 2.
Symmetric TSP is structurally equivalent to a constrained Group Steiner Tree Problem.
problem Finding the shortest tour in a symmetric TSP.
method Structural equivalence between symmetric TSP and constrained Group Steiner Tree Problem.
result Maximizing net weight in the cGSTP is equivalent to minimizing the TSP tour length.
Study stable equivalence relations on 4-manifolds, proving homotopy equivalent manifolds with abelian fundamental group are stably diffeomorphic.
problem Classifying stable equivalence relations on 4-manifolds.
method Combination of modified and classical surgery, focusing on homotopy equivalence up to stabilisation.
result Closed oriented homotopy equivalent 4-manifolds with abelian fundamental group are stably diffeomorphic.
A Lie algebroid classifies G-structures with connections.
problem Classifying G-structures with connections.
method Associate a Lie algebroid to G-structures with connections.
result The Lie algebroid encapsulates all equivalence information.
The paper compares isotopic and diffeomorphic links in lens spaces.
problem Comparing isotopic and diffeomorphic links in lens spaces.
method Provides a set of moves on disk, band, and grid diagrams to connect diffeomorphic links.
result There are up to four isotopy equivalent links in each diffeo equivalence class.
Universal connection constructed using diffeology theory.
problem Natural connection on bundles of paths on manifolds.
method Diffeological construction of Singer's universal connection.
result Functorial equivalence between holonomy categories and diffeological bundle-connection pairs.
We give two constructions of surfaces in simply-connected 4-manifolds with non simply-connected complements. One is an iteration of the twisted rim surgery introduced by the first author. We also construct, for any group G satisfying some simple conditions, a simply-connected symplectic manifold containing a symplectic…
Ribbon knots with isomorphic quandles are stably equivalent.
problem Determining when ribbon knots with isomorphic quandles are equivalent.
method Using quandle isomorphism to show stable equivalence after finite connected sums.
result Ribbon knottings with isomorphic quandles are stably equivalent.
Moves connect multibranched surfaces with same neighborhoods.
problem Connecting multibranched surfaces with identical neighborhoods.
method Introduces moves to connect multibranched surfaces.
result Any two multibranched surfaces can be connected in finitely many steps.
Introduces canonical connections for sub-Riemannian manifolds with constant symbol.
problem Equivalence problem in sub-Riemannian geometry.
method Introduces canonical grading and compatible affine connection.
result Completely computed structures for contact manifolds of constant symbol.
New 4D shapes without spine found.
problem Existence of spines in simply-connected 4-manifolds.
method Heegaard Floer d invariants to find obstructions.
result Infinitely many 4-manifolds homotopy to S2 with no spine. The study introduces a new equivalence for Poisson modules on complex projective varieties.
problem Understanding the structure of Poisson modules on complex projective varieties with mild singularities.
method Introducing a weak concept of Morita equivalence in the birational context for Poisson modules.
result Poisson modules are classified into three types based on their properties.
We investigate the validity of the equivalence principle along paths in gravitational theories based on derivations of the tensor algebra over a differentiable manifold. We prove the existence of local bases, called normal, in which the components of the derivations vanish along arbitrary paths. All such bases are expl…
We investigate the concept of projective equivalence of connections in supergeometry. To this aim, we propose a definition for (super) geodesics on a supermanifold in which, as in the classical case, they are the projections of the integral curves of a vector field on the tangent bundle: the geodesic vector field assoc…
This study connects Gaussian processes and RKHS, bridging two machine learning communities.
problem Understanding the relationship between Gaussian processes and RKHS.
method Examining connections and equivalences in regression, interpolation, and other topics.
result Established the equivalence between Gaussian Hilbert space and RKHS.
We prove that the category of abelian gerbes with connection over a smooth manifold is equivalent to a certain category of principal bundles over the free loop space. These bundles are equipped with a connection and with a "fusion" product with respect to triples of paths. The equivalence is established by explicit fun…
Homotopy equivalent to spheres, free factor complex of a free group.
problem Understanding the homotopy type of free factor complexes.
method Proving homotopy equivalence and connectivity of relevant complexes.
result Homotopy equivalence of free factor complexes to spheres and other complexes.
Short note proves Lie algebroid deformation cohomology preservation under fibration conditions.
problem Understanding deformation cohomology of Lie algebroids under fibration conditions.
method Proves preservation of deformation cohomology up to degree k for Lie algebroids under certain fibration conditions.
result Deformation cohomology of Lie algebroids is preserved up to degree k under specified fibration conditions.
Extends connections on Lie groupoids, proving completeness conditions.
problem Existence and completeness of multiplicative connections on Lie groupoid fibrations.
method Introduces and investigates multiplicative Ehresmann connections on Lie groupoid fibrations.
result Conditions for completeness of multiplicative connections on Lie groupoid fibrations.
New equivalence relation on ribbon graphs connects to virtual links.
problem Understanding virtual links through ribbon graphs.
method Introducing a new equivalence relation on ribbon graphs.
result Correspondence between virtual links and ribbon graphs.
Convolutional layers can be mathematically equated to fully connected layers.
problem Understanding the equivalence between convolutional and fully connected layers for neural networks.
method Demonstrated that convolutional operations can be converted to matrix multiplication, showing equivalence.
result Convolutional layers and fully connected layers are mathematically equivalent in linear cases.
Lecture notes on geometric structures of 2nd order ODEs.
problem Understanding the geometry of second-order ordinary differential equations.
method Construction of Cartan connection, computation of invariants, recognition of symmetric equations.
result Constructive aspects of local equivalence problem for 2nd order ODEs.
In this short note we give an elementary proof of the fact that connections and their geometric parallel-transport counterpart are equivalent notions.
It is emphasized that equivalent definitions of connections on modules over commutative rings are not so in noncommutative geometry.
We present a simple approach to questions of topological orbit equivalence for actions of countable groups on topological and smooth manifolds. For example, for any action of a countable group Γ on a topological manifold where the fixed sets for any element are contained in codimension two submanifolds, every orbit e…
If the fundamental group of the complement of a smooth embedding f: S^2 \subset R^4 is a cyclic group, the map can be deformed to the standard embedding by a generic one-parameter family with at most cusp singularities. If two smooth embeddings are connected by such a deformation, they will be called cusp equivalent. W…