Atiyah and Todd classes of Lie algebroids respect their Atiyah sequence.
problem Understanding the Atiyah and Todd classes of Lie algebroids.
method Analyzing the Atiyah sequence of Lie algebroids and proving class restrictions.
result Atiyah and Todd classes of dg manifolds arising from regular Lie algebroids respect the Atiyah sequence.
Study connections on Lie groupoids and stacks using Atiyah sequences.
problem No specific problem stated; general connections on Lie groupoids and stacks.
method Construct connections using Atiyah sequences associated with transversal tangential distributions.
result Detailed study and construction of connections on Lie groupoids and stacks.
Derives Atiyah sequence for noncommutative bundles.
problem Deciding when ∗-automorphisms lift to compatible ones. method Derivation-based Atiyah sequence derivation.
result Validates existence of compatible lifts.
Cohomological and homological spectral sequences are shown to be isomorphic.
problem Cohomological and homological Atiyah-Hirzebruch spectral sequences are not always isomorphic.
method Spanier-Whitehead duality is used to establish an isomorphism between the two spectral sequences.
result Cohomological and homological Atiyah-Hirzebruch spectral sequences are isomorphic for finite spectra.
Established equivalence of Atiyah classes for generalized holomorphic vector bundles.
problem Defining and comparing Atiyah classes for generalized holomorphic vector bundles.
method Used three approaches: \(\check{C}\)ech cohomology, first jet short exact sequence, and Lie algebroid pairs.
result Equivalence of Atiyah classes defined by different methods.
Paper develops a unified framework for Lie algebroid connections on various bundles.
problem Unified framework for Lie algebroid connections on vector and principal bundles.
method Generalized Atiyah algebroid structure and its short exact sequence.
result Explicit constructions of Atiyah classes for Lie algebroid connections.
We construct connections and characteristic forms for principal bundles over groupoids and stacks in the differentiable, holomorphic and algebraic category using Atiyah sequences associated to transversal tangential distributions.
Paper introduces Atiyah sequence for Lie groupoids and studies gauge transformations.
problem Defining connections on principal 2-bundles over Lie groupoids.
method Introduced Atiyah sequence, defined strict and semi-strict connections, constructed gauge transformations.
result Existence criterion for connections on principal 2-bundles over proper, étale Lie groupoids.
Develops theory of d-holomorphic connections on Klein surfaces.
problem No specific problem stated; focuses on theory development.
method Constructs Atiyah exact sequence for d-holomorphic bundles and provides existence criterion.
result Established theory of d-holomorphic connections and existence criterion.
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.
Let T be a torus. We present an exact sequence relating the relative equivariant cohomologies of the skeletons of an equivariantly formal T-space. This sequence, which goes back to Atiyah and Bredon, generalizes the so-called Chang-Skjelbred lemma. As coefficients, we allow prime fields and subrings of the rationals, i…
Paper presents new cobordism sequences for singular maps.
problem Understanding cobordism groups of singular maps.
method Develops analogous exact sequences to classical ones.
result Positive answer to Szűcs' question on cobordisms of immersions.
Computes immersions of C2-projective spaces using K-theory.
problem Computing immersions of equivariant projective spaces.
method Geometric filtration and localized slice spectral sequence.
result Obtained equivariant analogue of James periodicity.
We construct the Atiyah-Hirzebruch spectral sequence (AHSS) for twisted differential generalized cohomology theories. This generalizes to the twisted setting the authors' corresponding earlier construction for differential cohomology theories, as well as to the differential setting the AHSS for twisted generalized coho…
The paper derives a formula for Lefschetz number of a geometric endomorphism.
problem Calculating the Lefschetz number for a singular foliation.
method Adapting the Atiyah-Bott theorem to a geometric endomorphism of a complex of LT-parallel sections. result A formula for the Lefschetz number of a geometric endomorphism.
Study knot spaces and Atiyah duality in spectral categories.
problem Understanding the space of embeddings of a circle into higher-dimensional manifolds.
method Develop a cosimplicial model using Atiyah duality and prove a comodule version.
result Compute knot spaces in low degrees and establish isomorphisms on fundamental groups.
In this paper, we construct a category of short exact sequences of vector bundles and prove that it is equivalent to the category of double vector bundles. Moreover, operations on double vector bundles can be transferred to operations on the corresponding short exact sequences. In particular, we study the duality theor…
Motivated by the Atiyah-Floer conjecture, we consider SO(3) Santi-self-dual instantons on the product of the real line and a three-manifold with cylindrical end. We prove a Gromov-Uhlenbeck type compactness theorem, namely, any sequence of such instantons with uniform energy bound has a subsequence converging to a ty…
We consider spectral sequences in smooth generalized cohomology theories, including differential generalized cohomology theories. The main differential spectral sequences will be of the Atiyah-Hirzebruch (AHSS) type, where we provide a filtration by the Cech resolution of smooth manifolds. This allows for systematic st…
Infinite volume found in the thick part of PSLn(R)-Hitchin-Riemann moduli space.
problem Proving infinite volume in the thick part of PSLn(R)-Hitchin-Riemann moduli space. method Employing Goldman flows and internal sequences to find an infinite series of subsets of identical volume.
result Infinite total Atiyah--Bott--Goldman volume for n>2. Study of BGG sequences on foliated manifolds with transverse parabolic geometry.
problem Analysis of BGG sequences on foliated manifolds with transverse parabolic structures.
method Filtered calculus and transversal index theory for filtered manifolds.
result Derived curved BGG sequences for foliated manifolds with transverse parabolic geometry.
Established a version of the Atiyah-Floer conjecture for SO(3)-bundles.
problem Atiyah-Floer conjecture for admissible SO(3)-bundles
method Adapted to admissible SO(3)-bundles
result A version of the Atiyah-Floer conjecture established
Atiyah reviewed holomorphic vector bundles and gauge theories.
problem Holomorphic vector bundles and gauge theories.
method Review of Atiyah's work from 1952-1990.
result Holomorphic vector bundles and gauge theories are interconnected.
In this paper we study the local description of spaces of forms on transitive Lie algebroids. We use this local description to introduce global structures like metrics, ∗-Hodge operation and integration along the algebraic part of the transitive Lie algebroid (its kernel). We construct a Čech-de Rham bicomplex wit…
Paper computes Atiyah class for DG manifolds of amplitude +1.
problem Computing the Atiyah class for DG manifolds of specific amplitude.
method Computed the Atiyah class by encoding the derived intersection of sections and zero sections of vector bundles.
result Atiyah class vanishes if and only if the intersection is clean.
The subject of this paper is strongly homotopy (SH) Lie algebras, also known as L∞-algebras. We extract an intrinsic character, the Atiyah class, which measures the nontriviality of an (SH) Lie algebra A when it is extended to L. In fact, given such an SH Lie pair (L,A), and any A-module E, there ass…
Atiyah-Singer theorem links math fields, predicts topological insights.
problem Understanding the interplay between analysis, geometry, and topology.
method Analyzes and generalizes topological invariants in differential geometry.
result Predicts the index of elliptic operators based on topology.
The cobordism group N(Mn) of codimension-one immersions in the n-manifold Mn has a natural filtration induced by any cellular decomposition. The problem addressed in this paper is the explicit computation of the graded group gr∗N(Mn). We introduce some new invariants for immersions enlightening the Atiyah-H…
Lattice formulation captures Atiyah-Patodi-Singer index.
problem Capturing index of Dirac operators in lattice gauge theory.
method Generalized spectral flow for non-product structure near boundaries.
result Correctly captures continuum index for small lattice spacings.
Constructs a triple on an Atiyah algebroid with connection.
problem Dynamics of systems on principal bundles and Atiyah algebroids.
method Constructs a Tulczyjew triple on a principal bundle with connection, then reduces to the Atiyah algebroid.
result Dynamics of systems on principal bundles and Atiyah algebroids are discussed and applied.
We provide new conditions for the Strong Atiyah conjecture to lift to finite group extensions. In particular, we show cocompact special groups satisfy these conditions, so the Strong Atiyah conjecture holds for virtually cocompact special groups.
We prove Atiyah's conjecture for two special types of configurations of N points in the three-dimensional Euclidean space. For one of these types, it is shown that the stronger conjecture of Atiyah and Sutcliffe is valid.
Proves a lattice version of the Atiyah-Singer index theorem.
problem Index problems of Wilson-Dirac operators on lattice approximations of manifolds.
method Formulates and proves a K-theoretic formula for an index-type invariant. result Main theorem gives a formula for an index-type invariant of operators on lattice approximations of closed integral affine manifolds.
We state and prove a condition under which the strong Atiyah Conjecture carries over to subgroups. Moreover, we show that if a group satisfies the (strong) Atiyah Conjecture then any quotient with finite kernel does.
We introduce a mathematician-friendly formulation of the physicist-friendly derivation of the Atiyah-Patodi-Singer index of our previous paper. Our viewpoint sheds some new light on the interplay among the Atiyah-Patodi-Singer boundary condition, domain-wall fermions, and edge modes.
Geometric framework for inverse problems using foliations and dual connections.
problem Reconstruction problems in inverse problems.
method Vaisman foliations and Atiyah--Molino sequences to induce transverse foliations and dual connections.
result Unique, path-independent reconstruction with vanishing torsion and curvature duality.
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.
For every Lie pair (L,A) of algebroids we construct a dg-manifold structure on the Z-graded manifold M=L[1]⊕L/A such that the inclusion ι:A[1]→M and the projection p:M→L[1] are morphisms of dg-manifolds. The vertical tangent bundle TpM then inherit…
Paper establishes a new formula for Atiyah-Patodi-Singer index using eta invariants.
problem Calculating the Atiyah-Patodi-Singer index without invertibility of boundary operator.
method Using an asymptotic gluing formula for eta invariants and a splitting principle.
result Formula expressing index in terms of eta invariants of domain-wall massive Dirac operators.
Study finite group actions on exotic aspherical space forms.
problem Classify finite group actions on M#Σ where M is a closed aspherical space form and Σ is an exotic n-sphere. method Combines geometric and topological rigidity results with smoothing theory and spectral sequence computations.
result Classification of free actions of finite groups on M#Σ when M is 7-dimensional. Given a holomorphic line bundle L on a compact complex torus A, there are two naturally associated holomorphic ΩA--torsors over A: one is constructed from the Atiyah exact sequence for L, and the other is constructed using the line bundle (p1∗L∗)⊗(α∗L), where α is the addition map on $A\times…
New Q-manifolds theory integrates Lie algebroids.
problem Integrating Lie algebroids over smooth manifolds.
method Introducing Q-groupoids and Q-bundles, proving Lie algebroids arise from Q-manifolds.
result Transitive Lie algebroids over second countable, smooth manifolds are integrated to locally trivial Q-groupoids.
In [Wu], the noncommutative Atiyah-Patodi-Singer index theorem was proved. In this paper, we extend this theorem to the equivariant case.
Gravitational instantons are constructed as superpositions of Atiyah-Hitchin and Taub-NUT geometries.
problem Constructing gravitational instantons from Atiyah-Hitchin and Taub-NUT geometries.
method A gluing construction that captures the superposition of moduli spaces of centred SU(2) monopoles and Taub-NUT manifolds.
result Gravitational instantons are explicitly shown to be superpositions of Atiyah-Hitchin and Taub-NUT geometries.
Researchers construct an index map for contact manifolds using K-theory.
problem Constructing an index for maximally hypoelliptic operators on contact manifolds.
method Using Higson's construction for symbol class in K-theory, they derive a series of maps whose induced map in K-theory is the Heisenberg Atiyah-Singer index map.
result Explicit construction of a series of maps leading to the Heisenberg Atiyah-Singer index map.
We present the details of our embedding proof of the Atiyah-Patodi-Singer index theorem for Dirac operators on manifolds with boundary.
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…
Constructs a model for differential KO-theory using Clifford modules.
problem Refining Atiyah and Singer's families index with differential structure.
method Builds a model using families of Clifford modules with superconnection.
result Affords a differential refinement of Atiyah and Singer's families index.