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.
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.
Proves an index theorem for foliations using spectral triples.
problem Proving an Atiyah L2 covering index theorem for foliations. method Symbol calculus for foliations and spectral triples.
result Induces the same map on K-theory for two types of spectral triples.
The paper identifies manifolds with harmonic complex structures.
problem Understanding harmonic almost complex structures on Riemannian manifolds.
method Analyzing the Atiyah-Hitchin-Singer and Eells-Salamon almost complex structures on twistor spaces.
result Characterizes manifolds with harmonic complex structures.
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.
The paper explores connections between dg manifolds and homotopy Lie algebras.
problem Understanding the relationship between dg manifolds and homotopy Lie algebras.
method Study of formal exponential maps, Atiyah classes, and Kapranov L-infinity algebras.
result Existence of formal exponential maps linked to vanishing of Atiyah classes.
Constructs a rigorous path integral for supersymmetric spin manifolds.
problem Defining a rigorous path integral for N=1/2 supersymmetry.
method Using differential forms and iterated integrals on loop spaces of compact spin manifolds.
result Provides a rigorous background for Atiyah-Singer index theorem proofs.
Analyzes semi-characteristics on specific manifolds, proving a vanishing theorem.
problem Analyzing semi-characteristics on certain manifolds.
method Combines assembly maps and Hodge theorem perspectives.
result Proves an Atiyah type vanishing theorem.
The paper proves abelian convexity theorems using Kempf-Ness functions.
problem Establishing convexity along orbits in general settings.
method Using Kempf-Ness functions to prove abelian convexity theorems.
result Short proofs of Atiyah-Guillemin-Sternberg theorem and abelian convexity for gradient maps.
The Atiyah-Singer Dirac operator changes smoothly with small boundary adjustments.
problem Smoothness of the Atiyah-Singer Dirac operator under local boundary perturbations.
method Riesz continuity demonstrated for operator under L∞ perturbations of local boundary conditions. result Lipschitz bound for the map depends on smoothness and curvature of boundary conditions.
Constructs Chern-Weil classes for Cartan geometries.
problem Defines characteristic classes for Cartan geometries.
method Defines a subalgebra of polynomials on the Atiyah algebroid of Q and a characteristic map. result Recover classical Chern-Weil map for specific cases.
Atiyah's formulation of what is nowadays called the convexity theorem of Atiyah-Guillemin-Sternberg has two parts: (a) the image of the moment map arising from a Hamiltonian action of a torus on a symplectic manifold is a convex polytope, and (b) all preimages of the moment map are connected. Part (a) was generalized b…
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.
Heegaard Floer homology duality applied to 4D mapping tori.
problem Computing Heegaard Floer invariants of 4D mapping tori.
method Applied Heegaard Floer homology and duality theory.
result Computed invariants of 4D mapping tori using Lefschetz numbers.
Maps between classifying spaces for certain groups are studied, with rational cohomology results.
problem Analyzing maps between classifying spaces for specific groups.
method Study of maps Θ between null-components of homomorphism and based map spaces. result Surjectivity of map Θ in rational cohomology for certain groups, and non-surjectivity for others. Study the moduli space of instanton bundles on G2 manifolds.
problem Understanding the moduli space of instanton bundles on G2 holonomy manifolds.
method Employing G2 cohomology, decompose the moduli space into bundle moduli and G2 structure moduli.
result The moduli space of instanton bundles on G2 manifolds decomposes into bundle moduli and G2 structure moduli.
Paper solves a 4D topology problem using Pin(2)-equivariant Mahowald invariants.
problem Existence of Pin(2)-equivariant stable maps between certain representation spheres.
method Analysis of Pin(2)-equivariant Mahowald invariants and use of cell diagrams, Atiyah-Hirzebruch spectral sequence.
result Proves a '10/8+4'-Theorem and solves the 11/8-Conjecture.
Study shows infinite volumes of moduli spaces for certain groups.
problem Infinite volumes of Hitchin-Riemann moduli spaces for specific groups.
method Employed Goldman flows to find infinite disjoint subsets of identical volume.
result Proved infinite Atiyah-Bott-Goldman covolume for mapping class group actions.
The Atiyah-Singer index theorem is a topological formula for the index of an elliptic differential operator. The topological index depends on a cohomology class that is constructed from the principal symbol of the operator. On contact manifolds, the important Fredholm operators are not elliptic, but hypoelliptic. Their…
A group action on a moduli space is shown to be faithful.
problem Injectivity of a homomorphism from mapping class group to symplectic mapping class group.
method Instanton Floer homology, Atiyah-Floer Conjecture, Heegaard Floer strategy.
result The homomorphism is injective for surfaces of genus at least 2.
A map between twistor spaces is defined based on metrics, showing holomorphicity under specific conditions.
problem Understanding the conditions under which a map between twistor spaces is holomorphic.
method Defining a diffeomorphism based on Riemannian metrics and analyzing its properties under different conditions.
result The map is holomorphic under specific conditions (conformal or homothetic metrics), with implications for the Atiyah-Hitchin-Singer and Eells-Salamon structures.
Let Mn be a closed, connected n-manifold. Let $\mtm$ denote the Thom spectrum of its stable normal bundle. A well known theorem of Atiyah states that $\mtm$ is homotopy equivalent to the Spanier-Whitehead dual of M with a disjoint basepoint, M+. This dual can be viewed as the function spectrum, F(M,S), whe…
Detecting exotic spheres involves analyzing framed configuration spaces.
problem Detecting exotic spheres through homotopy type of truncated Disc-presheaves.
method Using gluing results for Disc-presheaves, Atiyah duality, and computations of mapping class groups.
result Conditions for detecting exotic spheres through framed configuration spaces.
We prove that to every inclusion A↪L of Lie algebroids over the same base manifold M corresponds a Kapranov dg-manifold structure on A[1]⊕L/A, which is canonical up to isomorphism. As a consequence, Γ(Λ∙A∨⊗L/A) carries a canonical L∞[1] algebra structure whose una…
Let S be a closed surface of genus g >= 2 and z in S a marked point. We prove that the subgroup of the mapping class group Map(S,z) corresponding to the fundamental group pi_1(S,z) of the closed surface does not lift to the group of diffeomorphisms of S fixing z. As a corollary, we show that the Atiyah-Kodaira surface …
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.
We give a generalization of toric symplectic geometry to Poisson manifolds which are symplectic away from a collection of hypersurfaces forming a normal crossing configuration. We introduce the tropical momentum map, which takes values in a generalization of affine space called a log affine manifold. Using this momentu…
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.
New index formula connects numerical and K-theoretic indices.
problem Equivariant index for proper group actions on manifolds.
method Developed a trace on group conjugacy classes to relate numerical and K-theoretic indices. result Shows that numerical index equals K-theoretic index under certain conditions. The paper introduces Atiyah classes for SH Lie pairs and modules.
problem Measuring nontriviality of SH Lie algebras and their extensions.
method Introduces Atiyah classes and studies their properties and invariance.
result Atiyah classes induce graded Lie algebra structures and Lie algebra module structures.
Mathematician-friendly formulation of Atiyah-Patodi-Singer index.
problem Boundary conditions and edge modes in domain-wall fermions.
method Mathematician-friendly derivation of Atiyah-Patodi-Singer index.
result New insights into the interplay of boundary conditions, domain-wall fermions, and edge modes.
We establish the Thom isomorphism in twisted K-theory for any real vector bundle and develop the push-forward map in twisted K-theory for any differentiable proper map f:X→Y (not necessarily K-oriented). The push-forward map generalizes the push-forward map in ordinary K-theory for any K-oriented differentiable…
The paper studies connections on Lie groupoid pairs and their linearization.
problem Linearizing actions on homogeneous spaces of Lie groupoid pairs.
method Develops a theory of connections on equivariant principal bundles, associates an Atiyah class, and uses it to show linearization conditions.
result Necessary and sufficient conditions for linearizing dressing actions and monodromies.
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.
We give two generalizations of the Atiyah-Bott-Berline-Vergne localization theorem for the equivariant cohomology of a torus action: 1) replacing the torus action by a compact connected Lie group action, 2) replacing the manifold having a torus action by an equivariant map. This provides a systematic method for calcula…
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.
Study heat flow for gauged holomorphic maps over Kähler manifolds.
problem Understanding the moduli space of gauged holomorphic maps.
method Gradient flow approach inspired by Atiyah and Bott, heat flow method.
result Global existence of smooth solutions of gradient flow equations.
The Witten class is derived from equivariant cohomology of a conformal loop space.
problem Deriving the Witten class using equivariant cohomology.
method Applying equivariant localization formula to a conformal loop space.
result The Witten class is obtained from the conformal loop space.
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.
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.
In this note, we unveil homotopy-rich algebraic structures generated by the Atiyah classes relative to a Lie pair (L,A) of algebroids. In particular, we prove that the quotient L/A of such a pair admits an essentially canonical homotopy module structure over the Lie algebroid A, which we call Kapranov module.
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.
We prove a localization formula for group-valued equivariant de Rham cohomology of a compact G-manifold. This formula is a non-trivial generalization of the localization formula of Berline-Vergne and Atiyah-Bott for the usual equivariant de Rham cohomology. As an application, we obtain a version of the Duistermaat-Heck…
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.
Geometrically interprets Atiyah class vanishing for Lie pairs.
problem Atiyah class vanishing for Lie pairs and Lie algebroids.
method New characterisation of ideal systems and geometric interpretation.
result Atiyah class vanishes for Lie pairs with kernel of fibration.
The paper constructs structures for Lie pairs and their Atiyah classes.
problem Understanding structures of Lie pairs and their Atiyah classes.
method Constructs dg-manifolds and dg-Lie algebroids for Lie pairs, showing quasi-isomorphisms and Atiyah classes.
result Induces a quasi-isomorphism between dg-Lie algebroids and Atiyah classes of Lie pairs.
We give a proof of the cobordism invariance of the index of elliptic pseudodifferential operators on sigma-compact manifolds, where, in the non-compact case, the operators are assumed to be multiplication outside a compact set. We show that, if the principal symbol class of such an elliptic operator on the boundary of …