Abstract shows mapping between foliation characteristic classes.
problem Mapping characteristic classes of foliations.
method Losik's approach to Gelfand formal geometry and Crainic-Moerdijk's Čech-de~Rham cohomology.
result Map between characteristic classes is non-injective.
We show that recently constructed invariants of 3-dimensional manifolds and of hyperkaehler manifolds (L.Rozansky and E.Witten, hep-th/9612216) come from characteristic classes of foliations and from Gelfand-Fuks cohomology. In particular, any symplectic foliation gives invariants of 3-manifolds. Our preprint has many …
Solves index problem for curved BGG sequences in parabolic geometry.
problem Index theory of curved Bernstein-Gelfand-Gelfand sequences.
method Utilizes K-homology and noncommutative geometry.
result Solves the index problem for BGG-sequences on flat parabolic geometry.
Study on characteristic classes for codimension-one foliations, showing non-triviality under specific conditions.
problem Characterizing non-trivial characteristic classes for codimension-one foliations.
method Using Gelfand-Fuchs cohomology and dynamical properties of holonomy groups.
result Non-triviality of certain characteristic classes for Reeb foliations, contradicting classical theorems.
This paper is devoted to the study of geometric structures modeled on homogeneous spaces G/P, where G is a real or complex semisimple Lie group and P⊂G is a parabolic subgroup. We use methods from differential geometry and very elementary finite-dimensional representation theory to construct sequences of invar…
Study projective and almost conformally symplectic structures on manifolds.
problem Relations between projective and almost conformally symplectic structures.
method Single almost conformally symplectic connection with totally trace-free torsion.
result Generalizes Fedosov structures and encodes variability of connections in projective class.
Extends Gelfand duality to various geometric and analytical categories.
problem Generalizing Gelfand duality to different types of manifolds and bundles.
method Unified cohomological argument for manifolds and suitable classes of functions for bundles.
result Gelfand duality extended to real analytic and Stein manifolds, and to vector, affine, and jet bundles.
We give a simple construction of the Bernstein-Gelfand-Gelfand sequences of natural differential operators on a manifold equipped with a parabolic geometry. This method permits us to define the additional structure of a bilinear differential cup product on this sequence, satisfying a Leibniz rule up to curvature terms.…
Novel proof technique for Gelfand-Fuks cohomology.
problem Comparing sheaf-like data over manifold Cartesian powers.
method Local-to-global analysis through generalized good covers and factorization algebras.
result Unified approach to Gelfand-Fuks cohomology.
The amoebas associated to algebraic varieties are certain concave regions in the Euclidean space whose shape reminds biological amoebas. This term was formally introduced to Mathematics in 1994 by Gelfand, Kapranov and Zelevinski. Some traces of amoebas were appearing from time to time, even before the formal introduct…
Let Γ be a finite group acting linearly on a vector space V. We compute the Lie algebra cohomology of the Lie algebra of Γ-invariant formal vector fields on V. We use this computation to define characteristic classes for foliations on orbifolds.
New geometric method constructs singular Gelfand-Tsetlin modules.
problem Constructing 1-singular Gelfand-Tsetlin modules.
method Using complex geometry and universal ring Do with vector space S. result Obtained a construction of the universal 1-singular Gelfand-Tsetlin gln(C)-module. New differential complexes on symplectic manifolds.
problem Developing calculus on symplectic manifolds.
method Coupling a symplectic manifold to a vector bundle with a constrained curvature.
result Construction of new differential complexes.
Study Bernstein-Gelfand-Gelfand complexes on Lipschitz domains, computing cohomology and applying to elasticity models.
problem Cohomology of BGG complexes on bounded Lipschitz domains.
method Computes cohomology of conformal deformation and Hessian complexes in Sobolev spaces, allowing multiple input complexes.
result Establishes conformal Korn inequality and proposes generalizations of continuum models with microstructures.
Lecture notes on BGG complexes using Lie groups and algebras.
problem Constructing BGG complexes on open domains.
method Representation theory of semisimple Lie groups and Lie algebras.
result Introduction of BGG complexes with Lie group and algebra insights.
New finite element method for complex forms in any dimension.
problem Discretization of complex forms in arbitrary dimensions.
method Finite element discretization of ℓ-form-valued k-forms on triangulations. result Generalizes existing finite element methods for various tensor fields.
For smooth manifolds equipped with various geometric structures, we construct complexes that replace the de Rham complex in providing an alternative fine resolution of the sheaf of locally constant functions. In case that the geometric structure is that of a parabolic geometry, our complexes coincide with the Bernstein…
Symplectic structure found on projective structures on surfaces with boundary.
problem Deformation space of projective structures on surfaces with boundary.
method Natural symplectic structure on the space, integrating the Adler-Gelfand-Dikii-space of the boundary.
result Space is a Hamiltonian space for the symplectic groupoid.
We prove the conjecture by Feigin, Fuchs and Gelfand describing the Lie algebra cohomology of formal vector fields on an n-dimensional space with coefficients in symmetric powers of the coadjoint representation. We also compute the cohomology of the Lie algebra of formal vector fields that preserve a given flag at th…
The paper constructs a symplectic groupoid for a specific Poisson structure.
problem Integrating the Adler-Gelfand-Dikii Poisson structure on Lie groups.
method Constructing a symplectic groupoid Morita equivalent to the quasi-symplectic groupoid.
result The constructed symplectic groupoid is Morita equivalent to the quasi-symplectic groupoid.
The study extends hypoellipticity to filtered manifolds and applies it to BGG sequences.
problem Analyzing hypoellipticity on general filtered manifolds.
method Extending Rockland criterion to pseudodifferential calculus, constructing parametrix, generalizing BGG machinery.
result Generalized BGG sequences are Rockland in a graded sense.
Working over a pseudo-Riemannian manifold, for each vector bundle with connection we construct a sequence of three differential operators which is a complex (termed a Yang-Mills detour complex) if and only if the connection satisfies the full Yang-Mills equations. A special case is a complex controlling the deformation…
The study analyzes Dirac operators twisted by specific bundles, revealing their geometric and regularity properties.
problem Analyzing Dirac operators twisted by ramified Euclidean line bundles.
method Describes closed extensions of Dirac operators in terms of Gelfand-Robbin quotient, constructs geometric realizations, and develops an L2 regularity theory. result Geometric realizations of the Gelfand-Robbin quotient and an L2 regularity theory are constructed. CR Killing operator derived from tractor calculus for CR structures.
problem Analyzing CR structures and their deformations.
method Tractor calculus and BGG operators applied to compatible almost CR structures.
result CR Killing operator is a first BGG operator for the modified adjoint tractor connection.
Gelfand and MacPherson provided a new formula for calculating Pontrjagin classes.
problem Calculating Pontrjagin classes of differential manifolds.
method Introduced a local and combinatorial formula.
result Expanded and clarified the original formula by Gelfand and MacPherson.
We prove a cyclic cohomological analogue of Haefliger's van Est-type theorem for the groupoid of germs of diffeomorphisms of a manifold. The differentiable version of cyclic cohomology is associated to the algebra of transverse differential operators on that groupoid, which is shown to carry an intrinsic Hopf algebraic…
This is an expanded version of a series of two lectures given at the IMA summer program "Symmetries and Overdetermined Systems of Partial Differential Equations". The main part of the article describes the Riemannian version of the prolongation procedure for certain overdetermined system obtained recently in joint work…
We give a complete construction of the Bernstein-Gelfand-Gelfand complex on real or complex projective space using minimal ingredients.
The hemisphere rigidity theorem connects to the Gelfand problem, providing a precise value for the extremal parameter.
problem Finding the extremal parameter for a specific nonlinear equation on a hemisphere.
method Interpreting the hemisphere rigidity theorem within the context of the Gelfand problem and applying it to a fourth-order Gelfand problem.
result A precise value for the extremal parameter is derived for the Gelfand problem under certain conditions.
Researchers create exact sequences for isotropic 2-Grassmannian.
problem Constructing exact sequences for isotropic 2-Grassmannian.
method Using Penrose transform over a double fibration.
result The constructed sequences are analogues of the Bernstein-Gelfand-Gelfand resolutions.
We present the solution of a longstanding internal problem of noncommutative geometry, namely the computation of the index of a transversally elliptic operator on an arbitrary foliation. The new and crucial ingredient is a certain Hopf algebra associated to the transverse frame bundle. Its cyclic cohomology is defined …
New Poisson bracket connects to logarithmic manifolds.
problem Constructing a new Poisson bracket compatible with existing structures.
method Developed a new local Poisson bracket compatible with Adler-Gelfand-Dickey brackets, leading to a dispersionless limit.
result Leading term defines a logarithmic Dubrovin-Frobenius manifold.
In this paper I give new elementary proofs of basic results of Gelfand, Kapranov and Zelevinskywhich express discriminants and resultants in terms of determinants of direct images of Cayley-Koszul complexes of sheaves.
Let U(n) be the unitary group, and u(n)∗ the dual of its Lie algebra, equipped with the Kirillov Poisson structure. In their 1983 paper, Guillemin-Sternberg introduced a densely defined Hamiltonian action of a torus of dimension (n−1)n/2 on u(n)∗, with moment map given by the Gelfand-Zeitlin coordinates. A few …
New integral-geometric formulae derived from normal densities ring.
problem Smooth versions of BKK theorem.
method Algorithm based on ring of normal densities.
result Smooth versions of BKK theorem obtained.
For a real or complex semisimple Lie group G and two nested parabolic subgroups Q⊂P⊂G, we study parabolic geometries of type (G,Q). Associated to the group P, we introduce a class of relative natural bundles and relative tractor bundles and construct some basic invariant differential operators on …
Constructs BGG resolutions for symplectic case.
problem Exactness of BGG resolutions in singular infinitesimal characters.
method Penrose transform over Lagrangian Grassmannian.
result Exactness of constructed complex over big affine cell.
Foundations laid for formal manifolds in differential geometry.
problem No specific problem stated; focuses on formal manifolds.
method Introducing formal manifolds, developing their theory, and proving finite products.
result Established a fully faithful contravariant functor and finite products in the category of formal manifolds.
Defines symplectic sectional curvature and its properties.
problem Defines symplectic sectional curvature and its properties.
method Defines and characterizes symplectic sectional curvature in terms of curvature tensor and its covariant derivatives.
result Characterizes symplectic sectional curvature in terms of curvature tensor and its covariant derivatives.
In earlier joint work with A. Connes on transverse index theory on foliations, cyclic cohomology adapted to Hopf algebras has emerged as a decisive tool in deciphering the total index class of the hypoelliptic signature operator. We have found a Hopf algebra H(n), playing the role of a `quantum structure group' for the…
Report on formalizing differential geometry in Lean.
problem Formalizing differential geometry in a proof assistant.
method Lean's type theory approach to formalization.
result Surprising differences between formal and informal proofs.
Derives localization formulas in Batalin-Vilkovisky formalism.
problem Localization in Batalin-Vilkovisky formalism.
method Equivariant localization formulas in Batalin-Vilkovisky formalism.
result Derives localization formulas in Batalin-Vilkovisky formalism.
Generalizes Gelfand's spectrum to monogenic spinor fields on compact Riemannian manifolds.
problem Extend Gelfand's spectrum concept to non-commutative spaces of spinor fields.
method Use Clifford algebras and monogenic spinor fields, proving essential lemmas and a Stone-Weierstrass theorem.
result Spectrum of monogenic spinor fields on compact Riemannian manifolds is homeomorphic to the manifold itself.
Formalizes synthetic differential geometry in Lean.
problem Formalizing synthetic differential geometry in a proof assistant.
method Formalization of synthetic differential geometry with Lean and mathlib.
result Proves a Taylor theorem for functions of several variables.
The study examines differential smoothness in specific Artin-Schelter regular algebras of dimension 5.
problem Investigating the differential smoothness of Artin-Schelter regular algebras of dimension 5.
method Analyzing the relationship between the number of generators and Gelfand-Kirillov dimension to identify structural obstructions.
result Certain two- and four-generator AS-regular algebras of global dimension five fail to admit a differential calculus, while a five-generator graded Clifford algebra provides a positive example.
Study of nonlinear PDEs using derived geometry and BV formalism.
problem Understanding non-linear PDEs via derived geometric methods.
method Derived enhancement of de Rham complex, algebro-geometric techniques, BV formalism.
result Natural derived enhancement of de Rham complex for nonlinear PDEs.
In this paper we provide some stability criteria for systems of linear subspaces of V⊗W and for systems of quotient coherent sheaves, using, respectively, the Hilbert-Mumford numerical criterion and moment map. Along the way, we generalize the Gelfand-MacPherson correspondence [11] from point sets to sets of …
Introduces a new geometric framework for non-perturbative BV-theory.
problem Non-perturbative generalization of BV-theory in infinite-dimensional spaces.
method Derived differential geometry and homotopical algebraic geometry.
result Concrete model of derived smooth stacks for encoding non-perturbative BV-theory.