This paper constructs a functor preserving unobstructedness in symplectic geometry.
problem Generalizing unobstructedness in symplectic geometry for arbitrary manifolds and Lagrangian submanifolds.
method Using filtered A-infinity categories and Lagrangian Floer theory, the paper constructs a 2-functor.
result The geometric transformation preserves unobstructedness of Lagrangian Floer theory.
Study character varieties of tangles to map immersed curves in the pillowcase.
problem Characterizing holonomy-perturbed traceless SU(2) character varieties.
method Examining marked tangles as endomorphisms in the cobordism category and using holonomy-perturbed traceless character variety functor.
result Endomorphisms of immersed curves in the pillowcase have the same image.
Geometrically generates Fukaya categories of Weinstein manifolds.
problem Generating Fukaya categories of Weinstein manifolds.
method Geometric approach using surgery formula and Floer cohomology.
result Wrapped Fukaya category generated by index n critical points.
Defines a new category structure on Weinstein manifolds using h-principle.
problem Defining a category structure on Weinstein manifolds.
method Defines a co/sheaf of categories on the skeleton of a Weinstein manifold using h-principle.
result Existence and uniqueness of the co/sheaf of categories up to isotopy.
Study shows how tangle moduli spaces relate to boundary surfaces.
problem Relating tangle moduli spaces to boundary surfaces.
method Use of composition in the Weinstein category and SU(2) holonomy perturbations.
result Holonomy perturbed SU(2) traceless flat moduli space is a Lagrangian immersion.
We introduce mappings between spaces of functions on (super)manifolds that generalize pullbacks with respect to smooth maps but are, in general, nonlinear (actually, formal). The construction is based on canonical relations and generating functions. (The underlying structure is a formal category, which is a "thickening…
Study on singularities of Lagrangian immersions with applications in Floer theory.
problem Understanding singularities of Lagrangian immersions.
method Applying Hamiltonian isotopy in the Weinstein tubular neighbourhood to express singular points as fold points with cusp points.
result Local expression of singular points of Lagrangian immersions as fold points with cusp points.
Paper proves equivalence of two Floer theories using pearly trees and Hamiltonian flows.
problem Proving equivalence between two Floer theories.
method Using pearly tree discs and local Hamiltonian flows.
result Equivalence relation between immersed Lagrangian Floer theory and Hamiltonian immersed Lagrangian Floer theory.
A symplectic groupoid G.:=(G1⇉G0) determines a Poisson structure on G0. In this case, we call G. a symplectic groupoid of the Poisson manifold G0. However, not every Poisson manifold M has such a symplectic groupoid. This keeps us away from some desirable goals: for example, establishing…
We link Ginzburg algebras to Weinstein manifolds and Legendrian knots.
problem Understanding the relationship between Ginzburg algebras and Weinstein manifolds.
method Associated a stopped Weinstein manifold to a quiver and subquiver, proving quasi-isomorphism of relative Ginzburg algebra and Chekanov-Eliashberg dg-algebra.
result Relative Ginzburg algebra is quasi-isomorphic to Chekanov-Eliashberg dg-algebra of a singular Legendrian unknot link.
This paper sets a lower limit for the size of Weinstein's Lagrangian tubular neighborhoods.
problem Finding the minimum size of Weinstein's Lagrangian tubular neighborhoods.
method Using the curvature tensor and second fundamental form of the submanifold.
result Explicit lower bounds for the radii of tubular neighborhoods are derived.
Diffeological submanifolds are a new type of submanifold in manifold theory.
problem Defining and understanding different types of submanifolds in manifold theory.
method Introducing diffeological submanifolds and comparing them with other types of submanifolds.
result A diffeological submanifold can be included in a manifold without being an immersion.
Proves nearby Lagrangian cocores are homotopically rigid in certain dimensions.
problem Homotopy rigidity of nearby Lagrangian cocores in Weinstein sectors.
method Spectral wrapped Donaldson-Fukaya category with orthogonal group coefficients.
result Inclusion followed by retract and quotient is null-homotopic.
Proves conditions for nearby special Lagrangians in Calabi-Yau manifolds.
problem Conditions for nearby special Lagrangians in Calabi-Yau manifolds.
method Analyzes fundamental group, Floer cohomology, and special Lagrangian conditions.
result Conditions for nearby special Lagrangians in Calabi-Yau manifolds.
Geometric interpretation of 3-manifold invariants using immersed curves.
problem Obstructing smooth equivalences between 4-manifolds and surfaces with boundary.
method Relating morphisms between bordered Floer invariants to cobordism maps via immersed curves in the punctured torus.
result Morphisms between immersed curve invariants compute certain cobordism maps.
Defines de Rham relative cotangent complex in tangent categories.
problem Characterizing immersions, submersions, local diffeomorphisms, and unramified morphisms in tangent categories.
method Systematic study of morphisms and their interactions, using algebraic geometry, differential geometry, and Cartesian differential categories.
result Defines de Rham relative cotangent complex in an arbitrary tangent category.
The paper is a summary of the results of the authors concerning computations of symplectic invariants of Weinstein manifolds and contains some examples and applications. Proofs are sketched. The detailed proofs will appear in our forthcoming paper. In the Appendix written by S. Ganatra and M. Maydanskiy it is shown tha…
Study SYZ transforms for immersed Lagrangian multi-sections in symplectic geometry.
problem Understanding the geometry of SYZ transforms on Lagrangian torus fibrations.
method Investigation of Lagrangian surgery and extension of holomorphic vector bundles via SYZ transform for immersed Lagrangian multi-sections.
result New equivalence in the immersed Fukaya category invariant under immersed Floer cohomology, providing a mirror to isomorphism of holomorphic vector bundles.
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.
During the last thirty years, symplectic or Marsden--Weinstein reduction has been a major tool in the construction of new symplectic manifolds and in the study of mechanical systems with symmetry. This procedure has been traditionally associated to the canonical action of a Lie group on a symplectic manifold, in the pr…
The article proves Lie's third theorem for Lie algebroids via singular Lie groupoids.
problem Lie's third theorem does not hold for Lie groupoids and Lie algebroids.
method Introducing a subcategory of diffeological spaces called quasi-etale, constructing a functor mapping singular Lie groupoids to Lie algebroids.
result Lie's third theorem is valid for Lie algebroids within the context of singular Lie groupoids.
The paper explores generic properties of minimal surfaces in high dimensions.
problem Understanding the behavior of minimal surfaces in high-dimensional spaces.
method Analyzing the space of conformal minimal immersions using Baire category theory.
result A generic conformal minimal immersion is chaotic in various ways, such as being non-proper, almost proper, and g-complete. New symplectic invariants linked to odd sphere bundles.
problem Understanding symplectic manifolds through Aoo-algebras.
method Equivalence of symplectic Aoo-algebras to de Rham algebras on sphere bundles.
result Symplectic Aoo-algebras can define intersection theory.
This paper constructs a Weinstein trisection for a surface bundle.
problem Constructing a trisection for a surface bundle.
method Weinstein trisection approach applied to ΣgimesΣh. result Explicit construction of a Weinstein trisection for S2imesS2. Homological mirror symmetry proved for symmetric squares of punctured spheres.
problem Proving homological mirror symmetry for symmetric squares of punctured spheres.
method Constructed quasi-equivalences between wrapped Fukaya categories and derived categories of coherent sheaves, using categorical resolutions and localisation.
result Wrapped Fukaya category of symmetric square quasi-equivalent to coherent sheaves on a singular surface.
Introduces algorithm for Weinstein handlebodies of certain divisors.
problem Constructing Weinstein handlebodies for complements of smoothed toric divisors.
method Explicit coordinates and simple example for algorithm.
result Produces Weinstein Kirby diagrams for complements of smoothed toric divisors.
Study Weinstein structures on toric divisors' complements.
problem Understanding Weinstein structures on toric divisors' complements.
method Define a partially-centered condition on Delzant polytopes, develop an algorithm for Weinstein handlebody diagrams.
result Explicit Weinstein structures for complements of smoothed toric divisors.
We study Weinstein 4-manifolds which admit Lagrangian skeleta given by attaching disks to a surface along a collection of simple closed curves. In terms of the curves describing one such skeleton, we describe surgeries that preserve the ambient Weinstein manifold, but change the skeleton. The surgeries can be iterated …
Survey of Weinstein symplectic manifolds, old and new.
problem Understanding Weinstein symplectic manifolds.
method Survey of basic notions and results.
result Discussion of open problems in the field.
Generalizes symplectic reduction to cosymplectic groupoid actions.
problem Symplectic reduction for cosymplectic groupoid actions.
method Introduced cosymplectic groupoid actions and proved a theorem.
result Proved a theorem analogous to Mikami-Weinstein theorem.
Stabilized convex symplectic manifolds are equivalent to flexible Weinstein manifolds.
problem Understanding the equivalence between stabilized convex symplectic manifolds and flexible Weinstein manifolds.
method Analyzing the homotopy type and symplectic properties of the manifolds.
result Stabilized convex symplectic manifolds are symplectomorphic to flexible Weinstein manifolds.
We prove generic fibre of Painlevé moduli spaces are Weinstein handlebodies.
problem Understanding the generic fibre of Painlevé moduli spaces.
method Attach Weinstein handles to Stokes Legendrian.
result Generic fibre of Painlevé moduli spaces are Weinstein handlebodies.
Extends symplectic reduction and theorem to Lie algebroids.
problem Symplectic reduction and theorem for Lie algebroids.
method Extends Marsden-Weinstein reduction and Darboux-Moser-Weinstein theorems.
result Obtained coisotropic embedding theorem for symplectic Lie algebroids.
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.
New method to encode Weinstein 4-manifolds using multisections with divides.
problem Encoding and manipulating Weinstein 4-manifolds.
method Using multisection diagrams with divides and algorithms based on Kirby-Weinstein handle decompositions and PALFs.
result Definition and symplectic implementation of monodromy on multisections with divides.
Proves Weinstein conjecture for specific contact manifolds.
problem Proving the Weinstein conjecture for a specific class of contact manifolds.
method Introduced iterated planar Lefschetz fibrations and open book decompositions to prove the conjecture.
result Contact manifolds supporting iterated planar open book decompositions satisfy the Weinstein conjecture.
The study translates Weinstein structures to Legendrian handlebodies for symplectic topology.
problem Detecting flexibility and rigidity in Weinstein manifolds.
method Systematic recipe for translating Weinstein Lefschetz fibrations to Legendrian handlebodies.
result Verification of Stein deformation equivalence and existence of closed exact Lagrangian submanifolds.
We prove symplectic hypersurfaces in Weinstein domains and give obstructions for manifold boundaries.
problem Obstructions for 3-manifolds to bound Weinstein domains in certain symplectic 4-manifolds.
method Symplectic embedding and deformation techniques.
result Obstructions for 3-manifolds to bound Weinstein domains in rational surfaces.
New invariant defined for Weinstein domains, related to Kirby-Thompson's invariant.
problem Defining a symplectic invariant for Weinstein domains.
method Contact cut graph, Lefschetz fibrations, multisections with divides.
result Definition of Weinstein L-invariant and its relation to Kirby-Thompson's invariant. Develops Marsden-Meyer-Weinstein reduction for k-contact field theories.
problem None explicitly stated, but related to field theories and contact geometry.
method Marsden-Meyer-Weinstein reduction techniques applied to k-contact field theories. result Clarifies and corrects previous contact reduction literature.
Proves existence of regular Lagrangians via Weinstein Lefschetz fibrations.
problem Existence of regular Lagrangians.
method Weinstein Lefschetz fibrations with a hypothesis.
result Existence of regular Lagrangians can be characterized by Lefschetz fibrations.
Abstract Lie algebroids generalize Lie algebroids to abstract categories.
problem Generalizing Lie algebroids to abstract categories.
method Generalized differentiation procedure to groupoid objects in categories with tangent structures.
result Abstract Lie algebroids are defined and examples include various groupoids.
The paper introduces a geometric flow for Lagrangian submanifolds that preserves Hamiltonian isotopy.
problem Finding stationary solutions for Hamiltonian stationary Lagrangian submanifolds.
method Introducing a geometric flow that is a gradient flow for volume and corresponds to a fourth order strictly parabolic scalar equation.
result Established short-time existence, uniqueness, and higher order estimates for compact initial Lagrangian immersions with uniformly bounded second fundamental forms.
The paper extends Marsden-Weinstein reduction to mechanical presymplectic structures for time-dependent Hamiltonian systems.
problem Limitations of Marsden-Weinstein reduction for cosymplectic structures in time-dependent Hamiltonian systems.
method Developed Marsden-Weinstein reduction for mechanical presymplectic structures.
result Mechanical presymplectic structures provide a more suitable framework for time-dependent Hamiltonian systems than cosymplectic structures.
Extending work of Chen, we prove the Weinstein conjecture in dimension three for strongly fillable contact structures with either non-vanishing first Chern class or with strong and exact filling having non-trivial canonical bundle. This implies the Weinstein conjecture for certain Stein fillable contact structures obta…
Study non-Weinstein Liouville geometry via hyperbolic dynamics, proving rigidity results.
problem Characterize non-Weinstein Liouville geometry with persistent transverse skeleton.
method Anosov 3-flows, Liouville Interpolation Systems, non-singular partially hyperbolic flows, hyperbolic dynamics.
result Mitsumatsu's examples characterize 4D non-Weinstein Liouville geometry with 3D persistent transverse skeleton.
Lie algebroids can not always be integrated into Lie groupoids. We introduce a new object--``Weinstein groupoid'', which is a differentiable stack with groupoid-like axioms. With it, we have solved the integration problem of Lie algebroids. It turns out that every Weinstein groupoid has a Lie algebroid and every Lie al…
Constructs Lagrangian skeleta for curve singularities.
problem Understanding Lagrangian skeleta of curve singularities.
method Constructs closed arboreal Lagrangian skeleta associated to links of isolated plane curve singularities.
result Provides computations of Legendrian and Weinstein invariants.