Deformation quantization yields a new moment map on symplectic diffeomorphisms.
problem Formalizing moment maps on diffeomorphism groups of symplectic manifolds.
method Deformation quantization framework applied to extrmDiff0(M). result Obtained a deformation of the Donaldson moment map.
New flow connects symplectic maps to hyperKähler geometry.
problem Understanding symplectic maps and their geometry.
method Established a correspondence between symplectic diffeomorphisms and hyperKähler moment maps.
result Introduced a new flow, the modified moment map flow.
Deform moment map on symplectic connections using star product algebras.
problem Understanding symplectic connections and their deformations.
method Study vector bundle of Fedosov star product algebras, formal connection, curvature, and star product trace.
result Showed star product trace as a formal symplectic form and moment map.
The symplectic representation of mapping classes is not surjective for certain types of mapping classes.
problem The surjectivity of the symplectic representation of mapping classes, particularly pseudo-Anosov ones, is not always preserved.
method Explicit construction of symplectic matrices with a bi-Perron leading eigenvalue that cannot be represented by orientable pseudo-Anosov mapping classes.
result The symplectic representation of orientable pseudo-Anosov mapping classes is not surjective.
Symplectic solitons rigid if bounded, study shows.
problem Understanding symplectic translating solitons with bounded second fundamental form.
method Using complex phase map to prove rigidity.
result Symplectic translating solitons rigid without bounded second fundamental form assumption.
We study iteration maps of recurrence relations arising from mutation periodic quivers of arbitrary period. Combining tools from cluster algebra theory and (pre)symplectic geometry, we show that these cluster iteration maps can be reduced to symplectic maps on a lower dimensional submanifold, provided the matrix repres…
Study shows symplectic mapping groups of K3 surfaces are infinitely generated.
problem Understanding symplectic mapping class groups of K3 surfaces.
method Uses Kronheimer's approach and Seiberg-Witten invariants.
result Symplectic mapping class groups of many K3 surfaces are infinitely generated.
Reduces cotangent bundles using symplectic methods.
problem Symplectic reduction of cotangent bundles.
method Locally conformally symplectic reduction at regular values.
result Proves theorem analogous to symplectic setting.
The paper finds symplectic mapping class relations using pencil pairs.
problem Understanding symplectic mapping class relations in 4-manifolds.
method Applying classification results for Fano 3-folds and polarized K3 surfaces to a general methodology of finding pencil pairs.
result Symplectic mapping class relations between products of positive Dehn twists.
Ancient symplectic solutions to mean curvature flow are flat.
problem Understanding ancient solutions to mean curvature flow in symplectic geometry.
method Using a complex phase map to prove a Bernstein theorem.
result Ancient solutions to the symplectic mean curvature flow are flat.
The paper explores the dynamics of composite symplectic Dehn twists with nonuniform hyperbolicity.
problem Understanding the dynamics and properties of composite symplectic Dehn twists.
method Analyzing the form of nonuniform hyperbolicity, growth of Floer cohomology, and classification of symplectic mapping classes.
result Composite symplectic Dehn twists exhibit positive topological entropy and exponential growth in Floer cohomology.
After reviewing recent results on symplectic Lefschetz pencils and symplectic branched covers of CP^2, we describe a new construction of maps from symplectic manifolds of any dimension to CP^2 and the associated monodromy invariants. We also show that a dimensional induction process makes it possible to describe any co…
The paper extends a theorem about momentum maps to singular symplectic spaces.
problem Extending a theorem about momentum maps to singular symplectic spaces.
method Using integral affine stratification and equivariant locally trivial fibrations, the paper extends the linear variation theorem to singular values of the momentum map.
result Cohomology classes of symplectic forms on reduced spaces vary linearly within strata.
New relation found in 4D symplectic mapping class group.
problem Relation between Dehn twists in symplectic 4-manifolds.
method Holomorphic curve techniques, symplectic isotopy problem solution.
result Relation between two products of Dehn twists.
The paper defines conditions for a Riemannian structure on a symplectic quotient.
problem Existence of Riemannian structures on symplectic quotients.
method Analyzes conditions for existence given a Lie group action with equivariant momentum mapping.
result Determines conditions under which an induced Riemannian structure exists.
Study differential operators over maps and their applications in supermanifolds.
problem Understanding differential operators over smooth maps and their applications.
method Recall and study differential operators, formal ℏ-differential operators, pullbacks by thick morphisms, and quantization of symplectic micromorphisms. result Developed constructions and examples of differential operators over maps.
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.
In [GMPS] we proved that the moment map image of a b-symplectic toric manifold is a convex b-polytope. In this paper we obtain convexity results for the more general case of non-toric hamiltonian torus actions on b-symplectic manifolds. The modular weights of the action on the connected components of the exceptio…
Geometric quantization for symplectic maps via Toeplitz operators.
problem Quantization of symplectic maps and Witten's conjecture.
method Berezin-Toeplitz operators and holomorphic sections over Kähler manifolds.
result Established a semi-classical trace formula for quantum representations of mapping class groups.
We prove that Manolescu and Woodward's Symplectic Instanton homology, and its twisted versions are natural, and define maps associated to four dimensional cobordisms within this theory. This allows one to define representations of the mapping class group and the fundamental group of a 3-manifold, and to have a geometri…
The well-known fact that any genus g symplectic Lefschetz fibration X4→S2 is given by a word that is equal to the identity element in the mapping class group and each of whose elements is given by a positive Dehn twist, provides an intimate relationship between words in the mapping class group and 4-manif…
Convexity theorem for Hamiltonian actions on conformal symplectic manifolds.
problem Establishing convexity in conformal symplectic geometry.
method Proving a convexity theorem for moment maps under Lee type actions.
result Analog of Kirwan's convexity theorem in conformal symplectic geometry.
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…
The space of symplectic connections on a symplectic manifold is a symplectic affine space. M. Cahen and S. Gutt showed that the action of the group of Hamiltonian diffeomorphisms on this space is Hamiltonian and calculated the moment map. This is analogous to, but distinct from, the action of Hamiltonian diffeomorphism…
This paper is concerned with the rational symplectic field theory in the Floer case. For this observe that in the general geometric setup for symplectic field theory the contact manifolds can be replaced by mapping tori of symplectic manifolds with symplectomorphisms. While the cylindrical contact homology is given by …
Renormalized circle diffeos with breaks converge to Moebius maps with a symplectic structure.
problem Analyzing the renormalization of circle diffeomorphisms with breaks.
method Proving convergence to invariant piecewise Moebius maps and identifying the renormalization operator with a sub-action of the mapping class group.
result Renormalization identifies with a symplectic form preserved by the mapping class group.
Reduces LCS manifolds with symplectic actions, preserving conformal structure.
problem Reduction of LCS manifolds with symplectic actions.
method Locally conformally symplectic reduction procedure.
result Compatibility with locally conformally Kähler structure and contact reduction.
Introduces group-valued momentum maps for symplectic fiber bundles.
problem Determining momentum maps for non-classical actions of automorphism groups.
method Develops a general framework for group-valued momentum maps.
result Illustrates the power of group-valued momentum maps with various examples.
Given an integral symplectic manifold, we construct a family of "coherent state" maps into complex projective space. The maps are built from sections of the tensor powers of a hermitian line bundle whose curvature is a multiple of the symplectic form. We show that this family is an almost-complex version of the Kodiara…
Constructs symplectic surface bundles with positive signatures.
problem Symplectic surface bundles over surfaces with positive signatures.
method Constructs symplectic surface bundles with specific genera.
result Determines commutator lengths of new mapping classes.
A mapping bends Teichmüller spaces into character varieties, preserving symplectic structure.
problem Mapping Fricke-Teichmüller space to character variety of surface representations.
method Bending Fuchsian representations along a fixed measured lamination, proving equivariant symplectic embedding and properness.
result Continuous extension of bending map to Thurston boundary and geometric complexification.
Compute group cohomology for mapping class group with non-symplectic coefficients.
problem Compute group cohomology for mapping class group with non-symplectic coefficients.
method Compute the invariant subspace of the rational group ring of a surface, truncated by powers of the augmentation ideal, under the action of the mapping class group.
result First group cohomology computation for the mapping class group with non-symplectic coefficients.
Theory of symplectic reduction in infinite dimensions developed.
problem Challenges in symplectic reduction in infinite dimensions.
method Normal form of momentum map for infinite-dimensional equivariant maps.
result Theory of singular symplectic reduction in infinite dimensions.
The paper connects moment maps to the stability of holomorphic fibrations.
problem Stability of holomorphic fibrations.
method Use of moment maps and K-stability criteria.
result Existence of optimal symplectic connections implies stability of fibrations.
The Pontryagin forms on 1-jet bundle of Riemannian metrics, are shown to provide, in a natural way, diffeomorphism-invariant pre-symplectic structures on the space of Riemannian metrics for dimensions n=4r−2. The equivariant Pontryagin forms provide canonical moment maps for these structures. In dimension two, the sy…
Study Dehn-Seidel twists on Lagrangian spheres in K3 surfaces.
problem Understanding configurations of Lagrangian spheres in symplectic K3 surfaces.
method Use Seiberg-Witten theory and tools from symplectic mapping class groups.
result Proves algebraic independence of certain twists and generation results.
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…
Goldman symplectic form and complex structure compatible on SL(3,R) Hitchin component.
problem Compatibility of Goldman's symplectic form with complex structure on SL(3,R) Hitchin component. method Proof of compatibility between Goldman's symplectic form and Labourie-Loftin complex structure.
result Goldman symplectic form and complex structure determine a pseudo-Kähler structure on SL(3,R) Hitchin component. We consider a connected symplectic manifold M acted on by a connected Lie group G in a Hamiltonian fashion. If G is compact, we prove give an Equivalence Theorem for the symplectic manifolds whose squared moment map ∥μ∥2 is constant. This result works also in the almost-Kähler setting. Then we…
Improved non-squeezing theorem for calibrated geometries proved.
problem Proving an improved non-squeezing theorem for calibrated geometries.
method Two proofs: direct and reduction to classical case.
result Established an improved non-squeezing theorem for calibrated geometries.
New computations show symplectic groups and mapping class groups have different properties regarding torsion.
problem Comparing properties of symplectic groups and mapping class groups.
method Using K-theory, Weil representations, and quantum representations. result Symplectic groups have uniformly bounded torsion, while mapping class groups have more complex torsion.
There exist three main approaches to reduction associated to canonical Lie group actions on a symplectic manifold, namely, foliation reduction, introduced by Cartan, Marsden-Weinstein reduction, and optimal reduction, introduced by the authors. When the action is free, proper, and admits a momentum map these three appr…
The paper explores stability properties of cohomology groups and norms in symplectic and mapping class groups.
problem Stability properties of bounded cohomology in mapping class groups and symplectic groups.
method Utilizes results from Bestvina and Fujiwara, calculates norms of signature classes, and estimates cohomology norms.
result The bounded cohomology of mapping class groups does not stabilize, while that of symplectic groups does not stabilize via isometries.
Teichmüller space realized as symplectic quotient.
problem Realizing Teichmüller space as a symplectic quotient.
method Infinite-dimensional symplectic manifold, volume-preserving diffeomorphisms, momentum map.
result Teichmüller space and moduli space realized as symplectic orbit reduced spaces.
In this paper we extend the results of Kirwan et alii on convexity properties of the moment map for Hamiltonian group actions, and on the connectedness of the fibers of the moment map, to the case of non-compact orbifolds. Our motivation is twofold. First, the category of orbifolds is important in symplectic geometry b…
Study obstructs symplectic structures on Mazur manifolds.
problem Obstructing symplectic structures on Mazur manifolds.
method Using Heegaard Floer homology cobordism maps.
result Obstructs the existence of symplectic structures on Mazur manifolds.
Any nontrivial homomorphism from the mapping class group of an orientable surface of genus g≥3 to $\GL(2g,\C)$ is conjugate to the standard symplectic representation. It is also shown that the mapping class group has no faithful linear representation in dimensions less than or equal to 3g−3.
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.