Introduces derived Lie n-groupoids with shifted symplectic structures.
problem Defines structures for higher groupoids and their symplectic properties.
method Introduced derived Lie n-groupoids and their shifted symplectic structures, defining shifted lagrangian structures and proving composition well-defined.
result Shows that the framework includes various reduction procedures.
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.
The paper classifies symplectic Lie and L∞ algebroids and their applications.
problem Classifying symplectic Lie and L∞ algebroids and their higher gauge symmetries. method Classifying zero-, one-, and two-shifted symplectic algebroids using classical geometric higher structures.
result New examples of twisted Courant algebroids from codimension-two cycles and symplectic interpretations of higher structures.
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.
Strictification of isotropic distributions on derived schemes with shifted symplectic forms.
problem Understanding isotropic distributions on derived schemes with specific symplectic structures.
method Proving strictification result for isotropic distributions on derived schemes equipped with negatively shifted homotopically closed 2-forms.
result Derived schemes with −2-shifted symplectic structures globally admit Lagrangian distributions. Lie algebroids linked to L∞ spaces in derived geometry.
problem Relating Lie algebroids to L∞ spaces in derived geometry. method Constructing a faithful functor from Lie algebroids to L∞ spaces and showing the relationship between representations and vector bundles. result Lie algebroids provide an essentially unique L∞ space, and a shifted-symplectic structure on a dg Lie algebroid produces a shifted-symplectic structure on the associated L∞ space. New contact structures defined on differentiable stacks.
problem Defining contact structures on differentiable stacks.
method Introducing 0-shifted and +1-shifted contact structures. result Shifted contact structures provide new insights into geometry.
The paper establishes a Lagrangian correspondence linking different geometric structures on complex varieties.
problem Identifying relationships between different geometric structures on complex varieties.
method Using perfect complexes and shifted symplectic geometries, the paper establishes a Lagrangian correspondence.
result A Lagrangian correspondence between shifted symplectic geometries of flat and Higgs perfect complexes.
The BPS decomposition theorem splits cohomology of symmetric stacks into invariant parts.
problem Decomposing the cohomology of smooth symmetric stacks into invariant parts.
method Using cohomological Hall induction and intersection cohomology of moduli spaces.
result Establishes the BPS decomposition theorem for various symplectic stacks.
Continues work on derived manifolds and symplectic schemes, constructing virtual classes.
problem Constructing virtual fundamental classes for derived manifolds and schemes.
method Cosection localization, reduced virtual fundamental classes, and applications to Donaldson-Thomas theory.
result Virtual fundamental classes for (−2)-shifted symplectic derived schemes are consistent with algebraic and differential geometric constructions. The paper integrates quasi-Poisson manifolds into multiplicative D-valued moment maps.
problem Integrating quasi-Poisson manifolds into a broader geometric framework.
method Develops new aspects of shifted symplectic and Poisson geometry, establishing Lie-type correspondences and systematic constructions.
result Identifies multiplicative D-valued moment maps integrating quasi-Poisson manifolds, extending known constructions.
This paper puts the theory of quasi-Hamiltonian reduction in the framework of shifted symplectic structures developed by Pantev, Toën, Vaquié and Vezzosi. We compute the symplectic structures on mapping stacks and show how the AKSZ topological field theory defined by Calaque allows one to neatly package the constructio…
Develops a new deformation theory for Dirac structures.
problem Interpolating between twisted Dirac and Poisson geometries.
method Introduces a new deformation theory compatible with Dirac geometry operations.
result Uniform deformation theory recovering various special cases.
Defines duals of higher vector bundles for Lie 2-groupoids.
problem Constructing duals for higher vector bundles over Lie 2-groupoids.
method Develops theory of n-duals for simplicial vector spaces, defines n-duals for Lie 2-groupoids, and studies their properties.
result Proposes a new construction for VB 2-duals of VB 2-groupoids, showing they are VB 2-groupoids themselves and have nondegenerate canonical dual pairings up to homotopy.
Let (X,ωX∗) be a separated, −2-shifted symplectic derived C-scheme, in the sense of Pantev, Toen, Vezzosi and Vaquie arXiv:1111.3209, of complex virtual dimension vdimCX=n∈Z, and Xan the underlying complex analytic topological space. We prove that …
We develop further the approach to derived differential geometry introduced in Costello's work on the Witten genus. In particular, we introduce several new examples of L-infinity spaces, discuss vector bundles and shifted symplectic structures on L-infinity spaces, and examine in some detail the example of derived loop…
Develops theory of differential graded schemes for derived stacks.
problem Creating a theory for derived stacks using dg schemes.
method Formulates dg schemes as homotopy sites, equates to stacks on dg algebras.
result Infinity category of stacks represented by dg schemes is derived schemes.
New concept of coisotropic structures for differentiable stacks defined.
problem Defining coisotropic structures for differentiable stacks.
method Using twisted Dirac structures and Morita equivalences.
result 1-shifted coisotropic structures transfer through Morita equivalences.
We prove that the Grothendieck-Springer simultaneous resolution viewed as a correspondence between the adjoint quotient of a Lie algebra and its maximal torus is Lagrangian in the sense of shifted symplectic structures. As Hamiltonian spaces can be interpreted as Lagrangians in the adjoint quotient, this allows one to …
Constructs cohomology decompositions for symmetric stacks.
problem Cohomology of symmetric stacks.
method Constructs decompositions of cohomology, Borel--Moore homology, and vanishing cycle cohomology.
result Defines BPS cohomology and proves its equivalence to intersection cohomology for smooth stacks.
This thesis extends contact structures to differentiable stacks using line bundle-valued 1-forms.
problem Extending classical contact structures to differentiable stacks.
method Introducing 0 and +1-shifted contact structures on Lie groupoids, using line bundle-valued 1-forms and homotopy kernels. result Definition and examples of 0 and +1-shifted contact structures on Lie groupoids. This work explores symplectic structures on graded manifolds and higher Lie groupoids.
problem Understanding symplectic structures on graded manifolds and their global counterparts.
method Introduction and study of graded manifolds, symplectic Q-manifolds, higher Lie groupoids, and their symplectic structures.
result Developed a graded analogue of Weinstein's tubular neighborhood theorem and explored its applications.
Survey of bundle gerbes in geometry, field theory, and quantization.
problem Exploring bundle gerbes and their applications in geometry, field theory, and quantization.
method Definition and classification of bundle gerbes with connection, surface holonomy, transgression line bundles, and geometric quantization.
result Bundle gerbes provide a smooth bordism-type field theory and geometric quantization for 2-plectic and symplectic forms.
This paper constructs cohomological Hall algebras for 3-Calabi-Yau categories.
problem Mathematical definition of the algebra of BPS states.
method Construction of cohomological Hall algebras for 3-Calabi-Yau categories.
result Construction of cohomological Hall algebras and proof of Joyce's conjecture.
The paper proves a new version of dimensional reduction in cohomological Donaldson-Thomas theory.
problem Proving a new version of dimensional reduction in cohomological Donaldson-Thomas theory.
method Using cohomological Donaldson-Thomas theory and loop stacks of 0-shifted symplectic stacks.
result Shows the BPS cohomology of loop stacks admits a description analogous to orbifold cohomology.
Introduces a new method for symplectic reduction along submanifolds.
problem Symplectic reduction in various geometric categories.
method Uniform approach to symplectic reduction in smooth manifolds, complex analytic spaces, and algebraic varieties.
result Generalizes and encompasses various known reduction techniques.
The paper extends orientability results for DT invariants on Calabi-Yau 4-folds.
problem Defining Donaldson-Thomas invariants on quasi-projective Calabi-Yau 4-folds.
method Extending orientability results to quasi-projective Calabi-Yau 4-folds and using gauge theory.
result Orientability of moduli stacks and invariants on quasi-projective Calabi-Yau 4-folds.
Defines compatibility between Riemannian and Jacobi structures.
problem Understanding the relationship between Riemannian and Jacobi structures.
method Introducing a notion of compatibility and proving results for specific examples.
result Establishes compatibility for Poisson, contact, and locally conformally symplectic structures.
The paper introduces new structures for left-symmetric algebroids.
problem Developing new mathematical structures for left-symmetric algebroids.
method Introducing Koszul-Vinberg-Nijenhuis structures and related concepts.
result Koszul-Vinberg-Nijenhuis structures provide a hierarchy of structures.
Defines compatibility between Riemannian and Jacobi structures.
problem Understanding compatibility between Riemannian and Jacobi structures.
method Introducing a notion of compatibility and proving it for specific structures.
result Fundamental examples of Jacobi structures lead to known structures like Riemann-Poisson, Kenmotsu, and locally conformally Kähler.
Extending Jacobi and Riemannian compatibility to Lie algebroids.
problem Generalizing compatibility between Jacobi and Riemannian structures.
method Generalizing previous work on fundamental examples to Lie algebroids.
result Compatibility results for Lie algebroids.
Study projective and direct limits of Banach structures with connections to G-structures.
problem Understanding connections between Banach structures and G-structures. method Endow projective and direct limits with Fréchet or convenient structures and study connections.
result Illustrated examples demonstrate the study of projective and direct limits.
Defines structure constants for specific geometric structures on Lie groups.
problem No specific problem stated; focuses on defining structure constants.
method Not explicitly detailed in the abstract.
result Defines structure constants for almost complex, almost symplectic, and Riemannian structures on a local Lie group.
Study on G2∗ structures and almost para-contact structures in 7D.
problem Understanding the relation between G2∗ structures and almost para-contact structures. method Calculating projections using properties of G2∗ structures. result Determined the class of almost para-contact structures induced by G2∗ structures. Defines a new Poisson structure for generalized Sasakian spaces.
problem No specific problem stated; focuses on new structure definition.
method Defines a canonical Poisson structure on generalized contact metric spaces.
result Shows distinction between generalized Sasakian and coKähler structures.
Study on types of generalized hypercomplex structures on tori and Kodaira-Thurston surface.
problem Characterizing types of generalized hypercomplex structures.
method Analysis of S2-family of generalized complex structures and study of twistor spaces. result Existence of generalized hypercomplex structures on 4n-dimensional tori with non-maximal types. Classifies complex Dirac structures with invariants and local structure.
problem Classifying complex Dirac structures.
method Introducing invariants (order, type), proving existence and splitting theorems.
result Pointwise classification and local structure of complex Dirac structures.
Study G2 structures on manifolds to find specific almost contact metric structures.
problem Understanding G2 structures and their induced almost contact metric structures. method Analyzing 2-fold vector cross products and their effects on manifolds.
result Identified possible classes of induced almost contact metric structures.
This article connects PCS structures to parabolic contact structures.
problem Connecting PCS structures to parabolic contact structures.
method Developed a parabolic version of contactification to show any PCS-structure can be locally realized.
result Any PCS-structure can be locally realized uniquely up to isomorphism in terms of parabolic contact structures.
Bihamiltonian structures lead to tau structures for integrable hierarchies.
problem Classifying deformations of bihamiltonian structures.
method Starting from flat exact semisimple bihamiltonian structures, we derive Frobenius manifolds and tau structures.
result Deformations of the principal hierarchy with tau structures are classified.
The paper explores geometric structures on Hom-Lie groups and algebras.
problem Exploring Kähler-Norden structures on Hom-Lie groups and algebras.
method Analyzing the relationship between holomorphic Norden structures and Kähler-Norden structures on Hom-Lie groups.
result Left-invariant holomorphic Hom-Lie groups with abelian complex structures are flat.
Extends corner structure study to general case, constructs normal Trans-Sasakian structures.
problem Extending corner structure study to general case without conditions.
method Extends corner structure to general case, constructs Trans-Sasakian structures from non-normal corner structures.
result Constructs normal Trans-Sasakian structures from non-normal corner structures.
Study on submanifolds in metallic structures with new results and structures.
problem Investigating submanifolds in metallic structures.
method Analyzing hypersurfaces and products spaces, defining new structures, and expressing fundamental theorems.
result New fundamental theorems for submanifolds in metallic structures.
New metric structures generalize Sasakian and cosymplectic structures, proving rigidity and finding conditions.
problem Generalizing Sasakian and cosymplectic structures to new metric structures.
method Introducing weak structures and proving rigidity of Sasakian structures.
result Any weak Sasakian structure is homothetically equivalent to a Sasakian structure.
Study GL(2)-structures on manifolds leading to complex structures.
problem Understanding GL(2)-structures and their relation to complex structures. method Explored GL(2)-structures on differential manifolds, proving their relation to almost-complex structures and providing a canonical connection. result Established a twistor-like construction for GL(2)-geometry. 3D projective structures can be metrized with conformal structures.
problem Weyl metrizability of 3D projective structures.
method Interpreting Weyl metrizability as CR submanifolds in 7D.
result Beltrami's theorem extends to conformal structures in 3D.
Hypersymplectic structures with torsion on Lie algebroids are investigated. We show that each hypersymplectic structure with torsion on a Lie algebroid determines three Nijenhuis morphisms. From a contravariant point of view, these structures are twisted Poisson structures. We prove the existence of a one-to-one corres…
Spin-harmonic structures on low-dimensional manifolds.
problem Defining geometric structures on low-dimensional manifolds.
method Introducing spin-harmonic structures defined by harmonic unitary spinors.
result Spin-harmonic structures are equivalent to balanced Spin(7) structures in dimension 8.