New 1-cocycle on homology cobordisms connects traces and Magnus representation.
problem Understanding traces and homology cobordisms through a new 1-cocycle.
method Explicit construction of a new 1-cocycle with properties connecting traces and Magnus representation.
result Rational abelianization of homology cobordism group is non-trivial.
New construction of Turaev-Viro invariants invariant under Morita equivalence.
problem Constructing Turaev-Viro invariants invariant under Morita equivalence.
method Pivotal bicategory construction of spherical module categories.
result The invariant recovers the standard Turaev-Viro invariant and is independent of the skeleton.
New parametrization of 3-spheres using Johnson subgroups.
problem Constructing integral homology 3-spheres.
method Intrinsic description of equivalence relation on fourth Johnson subgroup.
result Intrinsic description of equivalence relation using fourth Johnson handlebody subgroups.
Study subgroup of mapping class group related to handlebody, answering a question about Johnson homomorphism.
problem Understanding the image of the second Johnson homomorphism for a specific subgroup of mapping class groups.
method Introduced trace-like operators and used them to compute images of Johnson homomorphisms.
result Answered a question about algebraic description of the image of the second Johnson homomorphism.
Establishes Morita equivalence for Nijenhuis structures and proves invariance of modular class.
problem Morita equivalence for Nijenhuis structures and Poisson-Nijenhuis manifolds.
method Global-to-infinitesimal correspondence using Lie functor and enhanced known equivalences.
result Modular class of Poisson-Nijenhuis manifolds is invariant under Morita equivalence.
New cohomology theory shows compact Lie group actions are Morita invariant.
problem Establishing Morita invariance for cohomology of compact Lie group actions.
method Using bibundles to transfer coefficient systems between Morita equivalent groupoids.
result Twisted Bredon-Illman cohomology is Morita invariant for compact Lie group actions.
Unified framework for Morita invariant cohomology of Lie groupoids.
problem Proving Morita invariance of cohomology theories for Lie groupoids.
method Viewing cohomology as sheaves of modules on the nerve of the groupoid and establishing criteria for Morita invariance.
result Established criteria for Morita invariant cohomology theories.
These notes discuss various aspect of the ``representation theory'' of Poisson manifolds, with focus on Morita equivalence and Picard groups. We give a brief introduction to Poisson geometry (including Dirac and twisted Poisson structures) and algebraic Morita theory before presenting the geometric Morita theory of Poi…
Motivated by deformation quantization, we introduced in an earlier work the notion of formal Morita equivalence in the category of ∗-algebras over a ring $\ring C$ which is the quadratic extension by $\im$ of an ordered ring $\ring R$. The goal of the present paper is twofold. First, we clarify the relationship betw…
Proof that m-shifted symplectic forms are preserved under Morita equivalence of Lie n-groupoids.
problem Consistent definition of symplectic structures on higher Lie groupoids under Morita equivalence.
method Rigorous proof of m-shifted symplectic forms preservation.
result m-shifted symplectic forms are preserved under Morita equivalence of Lie n-groupoids.
Linearizability of singular foliations is preserved under a specific equivalence relation.
problem Preserving properties of singular foliations under equivalence relations.
method Characterization of tubular neighborhood embeddings using Euler-like vector fields.
result Linearizability along a leaf is a Morita invariant.
Study new invariants for string links using Milnor and Morita-Milnor homomorphisms.
problem Developing invariants for string links.
method Define and study total Milnor and infinitesimal Morita-Milnor homomorphisms.
result Introduced new invariants for string links.
Study VB-groupoids and their Morita invariance, with applications to Poisson geometry.
problem Characterize VB-Morita maps and prove Morita invariance of VB-cohomology.
method Establish fundamental theorem of VB-Morita maps and use it to prove invariance.
result Derived category of VB-groupoids is Morita invariant, leading to VB-stacks.
Study noncommutative orbifolds using spectral triples and Morita equivalence.
problem Understanding noncommutative geometry of group actions on orbifolds.
method Developed Morita equivalence for crossed product spectral triples.
result Noncommutative orbifolds are Morita equivalence classes of spectral triples.
New Morita equivalence for diffeological groupoids defined.
problem Defining Morita equivalence for diffeological groupoids.
method Developed diffeological groupoid actions, -bundles, and -bibundles; introduced principality; defined Hilsum-Skandalis tensor product.
result Biprincipal bibundles are weakly invertible in the bicategory DiffBiBund.
Study of Morita equivalences in Lie groupoids and their symmetries.
problem Exploring Morita equivalences in Lie groupoids.
method Investigation of principal bibundles and Morita equivalences in Pfaffian groupoids.
result Introduction and analysis of Pfaffian Morita equivalence.
In \cite{KOT:MORITA}, Kotschick and Morita showed that the Gel'fand-Kalinin-Fuks class in $\ds \HGF{7}{2}{}{8}$ is decomposed as a product η∧ω of some leaf cohomology class η and a transverse symplectic class ω. In other words, the Kontsevich homomorphism $\dsω\wedge :\HGF{5}{2}{0}{10} \rightarrow\HGF{7}{2}…
Residue formula for a complex invariant on orbifolds.
problem Calculating Morita-Futaki-Bott invariant on orbifolds.
method Using Grothendieck's residues for holomorphic vector fields with isolated singularities.
result Proved a residue formula for the invariant.
New models for symplectic structures on classifying stacks.
problem Building models for symplectic structures on classifying stacks.
method Introducing m-shifted symplectic Lie n-groupoids and constructing explicit symplectic Morita equivalences. result Explicit symplectic Morita equivalences between models of the 2-shifted symplectic structure on classifying stacks.
Study localizes integrals at isolated degenerate zeros.
problem Localization of Futaki-Morita integrals at isolated degenerate zeros.
method Streamlined exposition in the spirit of Bott, localization procedure for a holomorphic vector field on CPn. result Essentially unique formula for Futaki-Morita integral invariants.
Using Kontsevich's identification of the homology of the Lie algebra l_infty with the cohomology of Out(F_r), Morita defined a sequence of 4k-dimensional classes mu_k in the unstable rational homology of Out(F_{2k+2}). He showed by a computer calculation that the first of these is non-trivial, so coincides with the uni…
In this note we give geometric formulations and proofs of three results of S. Morita. These results relate certain two dimensional cohomology classes of various moduli spaces of curves. We also give a geometric interpretation of a fourth result of Morita. One motivation of this work is to facilitate the application of …
Johnson has defined a surjective homomorphism from the Torelli subgroup of the mapping class group of the surface of genus g with one boundary component to ∧3H, the third exterior product of the homology of the surface. Morita then extended Johnson's homomorphism to a homomorphism from the entire mapping cla…
New bridge between diffeology and noncommutative geometry.
problem Connecting diffeology and noncommutative geometry.
method Embedding quasifolds into diffeology and associating C*-algebras.
result Morita classes of C*-algebras associated with diffeomorphic quasifolds.
We obtain a combinatorial formula for the Miller-Morita-Mumford classes for the mapping class group of punctured surfaces and prove Witten's conjecture that they are proportional to the dual to the Witten cycles. The proportionality constant is shown to be exactly as conjectured by Arbarello and Cornalba [J. Alg. Geom.…
Constructs parallel transport in higher gauge theory using principal 2-bundles.
problem Higher gauge theory and parallel transport in categorified principal bundles.
method Explicit construction of parallel transport for connections on principal 2-bundles.
result Proves constructions fit into axiomatic framework for categorified parallel transport.
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.
Solves open problem on Lie groupoids equivalence.
problem Whether Lie groupoids Morita equivalent are diffeologically Morita equivalent.
method Localisation of 2-categories, anafunctors, Lie groupoids, diffeological groupoids.
result Two Lie groupoids diffeologically Morita equivalent are Morita equivalent in the Lie sense.
We study gauge transformations of Dirac structures and the relationship between gauge and Morita equivalences of Poisson manifolds. We describe how the symplectic structure of a symplectic groupoid is affected by a gauge transformation of the Poisson structure on its identity section, and prove that gauge-equivalent in…
Proposes a topological framework to study modular invariants and related concepts.
problem Exploring modular invariants and related concepts in topological quantum field theory.
method Topological paradigm in alterfold topological quantum field theory.
result Establishes a novel integral identity for modular invariance across multiple Morita contexts.
We extend certain homomorphisms defined on the higher Torelli subgroups of the mapping class group to crossed homomorphisms defined on the entire mapping class group. In particular, for every k≥2, we construct a crossed homomorphism εk which extends Morita's homomorphism τ~k to the entire mapping clas…
This thesis explores geometric stacks and Poisson manifolds, proving new results in their classification and equivalence.
problem Classifying and understanding geometric stacks and Poisson manifolds.
method Rigorous proofs and new site constructions for geometric stacks and Poisson manifolds.
result Classification and equivalence results for b-symplectic manifolds.
Compute Picard groups of stable b-symplectic manifolds.
problem Classify stable b-symplectic manifolds up to Morita equivalence.
method Introduce discrete invariants to classify manifolds.
result Compute Picard groups of stable b-symplectic manifolds.
Abstract: Bijection strengthened to Morita equivalence integrating Poisson and Cartan-Dirac structures.
problem Bijection between coadjoint orbits and conjugacy classes.
method Morita equivalence of quasi-symplectic groupoids integrating Poisson and Cartan-Dirac structures.
result Strengthened bijection to Morita equivalence.
Study mapping class groups and their representations linking to algebraic structures.
problem Understanding projective representations of mapping class groups and their Morita classes.
method Defined projective representations of mapping class groups using modular fusion categories and analyzed their irreducibility.
result Irreducible representations imply unique Morita-class of simple algebras.
Any etale Lie groupoid G is completely determined by its associated convolution algebra C_c(G) equipped with the natural Hopf algebroid structure. We extend this result to the generalized morphisms between etale Lie groupoids: we show that any principal H-bundle P over G is uniquely determined by the associated C_c(G)-…
We answer a question posed by Morita concerning the non-triviality of certain secondary characteristic classes for surface bundles. In doing so we are naturally led to show that a form of Harer stability holds for surface diffeomorphism groups in homology of small degree.
In this paper, we generalize the notion of Serre fibration to the Morita category of topological groupoids and derive the associated long exact sequence of homotopy groups. We use this results for calculation of homotopy groups of various groupoids, such as the foliation groupoid of a Riemannian foliation.
The paper introduces SRFs and I-Poisson manifolds, linking foliations to Riemannian geometry.
problem Understanding singular foliations and their properties in Riemannian geometry.
method Adapting singular foliations to Riemannian metrics, defining Morita equivalence, and introducing I-Poisson manifolds.
result Morita equivalent SRFs have isomorphic leaf spaces as pseudo-metric spaces.
Evaluating the twisted Morita-Mumford classes bar(h)_p on the Artin braid group B_n, we give the stable algebraic independence of the bar(h)_p's on the automorphism group of the free group, Aut(F_n). This is sharper than the results we obtained by restricting them to the mapping class group.
Study K3 manifold bundles proving Miller--Morita--Mumford classes non-zero.
problem Proving non-zero characteristic classes for K3 manifold bundles.
method Two methods: Franke's stable cohomology result and Miller--Morita--Mumford classes.
result Proves non-zero characteristic classes for certain bundles.
Develops Johnson-Morita theory for 3D handlebody groups.
problem Algebraic structure of 3D handlebody groups.
method Analogue of Johnson-Morita theory for handlebody groups.
result Associated graded embeds into a Lie algebra of special derivations.
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.
An orbifold is a Morita equivalence class of a proper {\' e}tale Lie groupoid. A unitary equivalence class of spectral triples over the algebra of smooth invariant functions are associated with any compact spin orbifold. In the case of an effective spin orbifold we construct a collection of spectral triples over the sm…
Kotschick and Morita recently discovered factorisations of characteristic classes of transversally symplectic foliations that yield new characteristic classes in foliated cohomology. We describe an alternative construction of such factorisations and construct examples of topologically trivial foliated vector bundles fo…
Undecidability proved for DG algebras problems.
problem Stable isomorphism, quasi-isomorphism, and Morita equivalence problems for semifree DG algebras.
method Essentially autonomous solutions by Gemini Deep Think and Aletheia.
result Proved undecidability of problems for semifree DG algebras.
A new invariant of Poisson manifolds, a Poisson K-ring, is introduced. Hypothetically, this invariant is more tractable than such invariants as Poisson (co)homology. A version of this invariant is also defined for arbitrary algebroids. Basic properties of the Poisson K-ring are proved and the Poisson K-rings are calcul…
Study on smooth bundles with nonpositive curvature, proving vanishing classes and rigidity.
problem Characterizing smooth bundles with nonpositive curvature fibers.
method Proving vanishing results for Miller-Morita-Mumford classes and showing topological rigidity.
result Vanishing of generalized Miller-Morita-Mumford classes and topological rigidity of vertical tangent bundles.