Classifies discrete vector bundles over simplicial complexes, generalizing Weil's theorem.
problem Classification of discrete vector bundles over simplicial complexes.
method Discrete Differential Geometry approach, including classification theorem and curvature association.
result Discrete hermitian line bundles with curvature have a unique piecewise-smooth counterpart.
Develops combinatorial theory of vector bundles on simplicial complexes.
problem Creating a discrete theory for vector bundles and connections on simplicial complexes.
method Introduces discrete exterior covariant derivative and applies it to various geometric objects.
result Flat discrete connections yield a cochain complex computing twisted de Rham cohomology.
Study of parabolic vector bundles on Klein surfaces.
problem Understanding parabolic vector bundles on Klein surfaces.
method Defined and studied parabolic vector bundles, proving isomorphism classes correspond to representations of discrete subgroups.
result Isomorphism classes of polystable real and quaternionic parabolic vector bundles correspond to equivalence classes of representations of discrete subgroups.
Develops mixed quantization for graph vector bundles.
problem Solving asymptotic spectral problems on graph vector bundles.
method Mixed quantization technique for graph vector bundles.
result Applications to various spectral problems.
A new discrete calculus for bundle-valued forms is proposed and validated.
problem Discretization of exterior calculus for bundle-valued forms.
method Discretization of Cartan's exterior calculus for differential forms with values in vector bundles.
result The proposed discrete operator mimics the continuous exterior covariant derivative and ensures numerical convergence.
We establish a uniform comparison between the spectrum of the rough Laplacian (acting on sections of a vector bundle of complex rank one or of harmonic curvature) with the spectrum of a discrete operator (a generalization of a discrete magnetic Laplacian added with a potential) acting on a finite dimensional space comi…
Study of vector bundles over classifying spaces for infinite discrete groups.
problem Understanding vector bundles over classifying spaces for infinite discrete groups.
method Homotopy theoretical framework for infinite discrete groups, relating to Novikov conjecture.
result Established a connection between representation spaces and vector bundles over classifying spaces.
Defines discrete differential geometry concepts in homotopy type theory.
problem No existing definition of Euler characteristic for comparison.
method Type families on higher inductive types, simplicial complexes, principal bundles, connections, curvature, vector fields, index.
result Theorem relating total curvature and total index, key to proving Gauss-Bonnet and Poincaré-Hopf theorems.
We establish an asymptotic relation between the spectrum of the discrete Laplacian associated to discretizations of a half-translation surface with a flat unitary vector bundle and the spectrum of the Friedrichs extension of the Laplacian with von Neumann boundary conditions. As an interesting byproduct of our study, w…
Study of discrete analogues of Atiyah sequence in principal bundles.
problem Discrete analogues of vector bundles and connections in principal bundles.
method Analysis in two categories: fiber bundles with sections and local Lie groupoids, defining discrete curvature and splittings.
result Correspondence between splittings of discrete Atiyah sequence and discrete connections with trivial curvature.
Unified approach to data processing using gauge theory.
problem Data representation and analysis with consistent symmetry.
method Geometric gauge theory for discrete vector bundles.
result Unified understanding of heat kernel properties and data transformation.
We prove the existence of extremal, non-csc, Kähler metrics on certain unstable projectivised vector bundles ¶(E)→M over a cscK-manifold M with discrete holomorphic automorphism group, in certain adiabatic Kähler classes. In particular, the vector bundles E→M under consideration are assumed to split as a …
In 1980, I. Morrison proved that slope stability of a vector bundle of rank 2 over a compact Riemann surface implies Chow stability of the projectivization of the bundle with respect to certain polarizations. Using the notion of balanced metrics and recent work of Donaldson, Wang, and Phong-Sturm, we show that the stat…
Study asymptotic expansion of graph Laplacian on discretized surfaces, relating spanning trees and cycle-rooted forests.
problem Asymptotic expansion of graph Laplacian on discretized surfaces.
method Relate spanning trees and cycle-rooted spanning forests to zeta-regularized determinants.
result Explicit formula for limit of cycle-rooted spanning forest probability and topological observables.
In 1980, I. Morrison proved that slope stability of a vector bundle of rank 2 over a compact Riemann surface implies Chow stability of the projectivization of the bundle with respect to certain polarizations. We generalized Morrison's result to higher rank vector bundles over compact algebraic manifolds of arbitrary di…
Study iterations of evolutes and involutes of curves with singularities.
problem Analyzing the behavior of evolutes and involutes of curves with singularities.
method Discretization of evolutes and involutes using polygons and their circumcenters or incenters; study of dynamics of inverse constructions.
result Characterization of linear maps induced by evolutes and involutes on polygon spaces.
To every Hermitian vector bundle with connection over a compact Riemannian manifold M one can associate a corresponding connection Laplacian acting on the sections of the bundle. We define analogous combinatorial metric dependent Laplacians associated to triangulations of M and prove that their spectra converge, as…
In a way similar to the continuous case formally, we define in different but equivalent manners the difference discrete connection and curvature on discrete vector bundle over the regular lattice as base space. We deal with the difference operators as the discrete counterparts of the derivatives based upon the differen…
In this paper we study the analytic realisation of the discrete series representations for the group G=Sp(1,1) as a subspace of the space of square integrable sections in a homogeneous vector bundle over the symmetric space G/K:=Sp(1,1)/(Sp(1)×Sp(1)). We use the Szegö map to give expressions for the restric…
We study a discrete dynamical system designed to find a 'most holomorphic' connection on a smooth complex vector bundle E. We examine the relation between the distance of the chern classes of E from the (p,p) axis of the Hodge diamond and singularity formation. Canonical connections and canonical metrics pulled b…
Develop contact Tulczyjew formalism for dissipative dynamics on skew algebroids.
problem Dissipative dynamics on skew algebroids
method Contact Tulczyjew formalism
result Intrinsic explanation of contact term and Euler-Lagrange-Herglotz equations
Discrete connections on abelian Lie groups bundles are studied.
problem Understanding discrete connections on abelian Lie group principal bundles.
method Formalized discrete connections as singular cochains and proved a discrete holonomy formula.
result Discrete connections on abelian Lie group bundles have properties similar to continuous connections.
This work revisits, from a geometric perspective, the notion of discrete connection on a principal bundle, introduced by M. Leok, J. Marsden and A. Weinstein. It provides precise definitions of discrete connection, discrete connection form and discrete horizontal lift and studies some of their basic properties and rela…
Market bubbles identified via spectral theory of geometric bundles.
problem Identifying asset bubbles in markets with arbitrage opportunities.
method Geometric Arbitrage Theory reformulated as a stochastic principal fibre bundle with a connection Laplacian.
result A market satisfies (NFLVR) if and only if 0 is in the discrete spectrum of the connection Laplacian.
The paper defines connections on Lie groupoid bundles and their properties.
problem Defining connections on Lie groupoid bundles and their properties.
method Introducing a suitable definition for connections on Lie groupoid bundles and proving their existence.
result Existence of connections on Lie groupoid bundles and their properties.
Monograph introduces bounded cohomology of discrete groups and spaces.
problem Understanding the structure of discrete groups and topological spaces.
method Definition and study of bounded cohomology.
result Applications in simplicial volume, circle actions, and flat vector bundles.
Connections on principal bundles play a fundamental role in expressing the equations of motion for mechanical systems with symmetry in an intrinsic fashion. A discrete theory of connections on principal bundles is constructed by introducing the discrete analogue of the Atiyah sequence, with a connection corresponding t…
Analytic torsion form constructed for non-commutative spaces.
problem Constructing analytic torsion form on non-commutative spaces.
method Using fiber bundles, flat vector bundles, and crossed product algebras, we define a non-commutative deRham differential form.
result The constructed torsion form appears in a transgression formula and leads to an index formula.
The paper defines and analyzes n-fold vector bundles and their decompositions.
problem Understanding and decomposing n-fold vector bundles. method Introducing n-fold vector bundles as functors and studying their cores and decompositions. result Any n-fold vector bundle admits a non-canonical isomorphism to a decomposed n-fold vector bundle. The paper proves structures on vector bundles with free actions.
problem Understanding isometry structures on vector bundles.
method Analyzes vector bundles with free canonical isometric actions.
result Total space of vector bundles carries principal bundle structures.
Study of generalized vector bundles and their geometric tools.
problem Extension of differential geometric tools to infinite dimensional vector bundles.
method Analysis of automorphisms, frame bundle, connection 1-forms, and covariant derivatives in diffeological vector pseudo-bundles.
result Non-isomorphism between connection 1-forms and covariant derivatives in infinite dimensional cases.
Proves every equivariant vector bundle over toric manifolds is a Klyachko bundle.
problem Characterizing equivariant vector bundles over toric manifolds.
method Analyzes topological and smooth equivariant vector bundles over toric manifolds.
result Every equivariant vector bundle is a Klyachko bundle.
Study on involution in double jet bundles.
problem Exploring involution in double jet bundles.
method Generalized double tangent bundles to double jet bundles, presented secondary vector bundle structure, proved involution.
result Existence of a natural involution on double jet bundles.
Let G0 be a connected, simply connected real simple Lie group. Suppose that G0 has a compact Cartan subgroup T0, so it has discrete series representations. Relative to T0 there is a distinguished positive root system Δ+ for which there is a unique noncompact simple root ν, the "Borel -- de Siebenthal s…
The paper explores grids and warps in triple vector bundles, proving a zero-sum property and applying it to manifold and vector bundle contexts.
problem Understanding the structure and commutativity of triple vector bundles.
method Intrinsic proof of the sum of warps being zero, applied to specific cases like tangent bundles and vector bundles.
result The sum of warps in a triple vector bundle is zero, with applications to manifold and vector bundle structures.
Proves GAGA-style result for toric vector bundles.
problem None explicitly stated in the abstract.
method Algebraic construction of Frölicher approximating vector bundle.
result Proves GAGA-style result for toric vector bundles.
Constructs a functor for graded bundles to vector bundles, characterizing symmetric structures.
problem Characterizing and fully characterizing the image of a functor from graded bundles to vector bundles.
method Constructs a full linearisation functor that takes a graded bundle of degree k to a k-fold vector bundle, fully characterizing the image and discussing related cases.
result Obtains a subcategory of k-fold vector bundles consisting of symmetric k-fold vector bundles equipped with a family of morphisms indexed by the symmetric group Sk. The paper characterizes vector fields as interpolating sesqui-harmonic maps on Riemannian manifolds.
problem Characterizing vector fields as interpolating sesqui-harmonic maps on Riemannian manifolds.
method Characterization theorem and critical point condition for interpolating sesqui-harmonic vector fields.
result Conditions for vector fields to be interpolating sesqui-harmonic maps on compact manifolds.
Classifies equivariant vector bundles over toric manifolds.
problem Classifying vector bundles over toric manifolds.
method Klyachko-type classification over invariant affine charts.
result Generalizes Klyachko's classification of toric vector bundles.
This paper shows vector bundles and differential bundles are equivalent in smooth manifolds.
problem Characterizing vector bundles in smooth manifolds.
method Introducing differential bundles in a tangent category and proving equivalence with vector bundles in smooth manifolds.
result Differential bundles in smooth manifolds are equivalent to vector bundles.
New pushforward operation on vector pseudo-bundles creates new examples.
problem Creating new objects from vector bundle theory in diffeology.
method Introducing pushforward operation on diffeological vector pseudo-bundles.
result Pushforward operation produces new projective diffeological vector spaces.
Introduces new super vector bundles and discusses their properties.
problem Homotopy classification of vector bundles in supergeometry.
method Introduction of ν-grassmannians and Γ, construction of Gauss supermap.
result Generalization of a theorem in homotopy classification for supergeometry.
The study of quotient structures in multi-graded bundles, including double vector bundles.
problem Understanding quotients of multi-graded bundles, especially double vector bundles.
method Analyzing quotients as towers of affine bundles and constructing normal bundles.
result Any quotient of multi-graded bundles fits into a tower of affine bundles.
Exponential localization of eigensections for Bochner-Schrödinger operator.
problem Understanding spectral properties of Bochner-Schrödinger operator on high tensor powers of Hermitian line bundles.
method Approximation of operator by model Schrödinger operator with constant magnetic field, analysis of spectrum.
result Spectrum of Bochner-Schrödinger operator in gaps is discrete and eigensections decay exponentially.
Defines connections on parabolic vector bundles for Lie algebroids.
problem Characterizing parabolic vector bundles with Lie algebroid connections.
method Constructs Lie algebroid connections on parabolic vector bundles, uses Atiyah exact sequence.
result Characterizes stable Lie algebroid vector bundles with connections.
The paper extends positivity results from vector bundles to Kobayashi positive ones.
problem Extending positivity results from vector bundles to Kobayashi positive ones.
method Using convexity of Kobayashi positive Finsler metrics and duality for convex Finsler metrics.
result The quotient and tensor product of Kobayashi positive vector bundles are also Kobayashi positive.
Study on singularities of bundle homomorphisms induced by Morin maps.
problem Characterizing singular points of bundle homomorphisms.
method Analyzing conditions for singularities induced by Morin maps, using Hamilton vector fields for contact structures.
result Characterization of singularities in bundle homomorphisms induced by Morin maps.
Linearizes nonlinear connections on vector and affine bundles.
problem Defining linear connections on nonlinear bundles.
method Linearization procedure for vector fields.
result Applications in Classical Mechanics.