22 papers
This paper proves that for knots containing specific components like the -torus knot or the figure-eight knot, no compactly supported Hamiltonian diffeomorphism can move their conormal bundles to intersect the zero section cleanly along an unknot, a result established by deriving a unique algebraic constraint on the augmentation variety over the rational numbers using symplectic field theory.
This paper, based on C. Viterbo's 2025 CIME lectures, establishes fundamental properties of the completion of Lagrangian submanifolds under the spectral metric via the refined concept of -support and applies these results to conformally symplectic dynamics to generalize the notion of Birkhoff attractors.
This paper clarifies the abstract geometrical formulation of thermodynamics on non-compact symmetric spaces used in Cartan Neural Networks by proving that only Kähler spaces support Gibbs distributions, explicitly characterizing their generalized temperature spaces via adjoint orbits, and demonstrating the equivalence between various information and thermodynamical geometries while establishing the covariance of these distributions under the full symmetry group.
This paper establishes an almost equivalence between the equivariant Tamarkin category and the category of derived complete modules over the Novikov ring, thereby clarifying the relationship between Tamarkin's extra variable and Novikov rings.
This paper constructs an instanton cochain complex using the eigenspaces of a deformed mapping cone Laplacian and proves its cochain isomorphism to the topologically defined mapping cone Thom-Smale complex, thereby establishing a bridge between analytic and topological approaches to the wedge by a smooth closed form.
This paper demonstrates that for a symplectic manifold with a monoidal Fukaya category, the monoidal structure uniquely determines the homological mirror functor to the derived category of coherent sheaves on its mirror , thereby filling a gap in the literature regarding the reconstruction of the mirror from categorical data.
This paper introduces the framework of "fat Lie theory" to establish a one-to-one correspondence between fat extensions and abstract 2-term representations up to homotopy, while demonstrating that fat extensions of groupoids correspond to general linear PB-groupoids and core extensions to vertically/horizontally core-transitive double groupoids, thereby upgrading existing correspondences to equivalences of categories.
This paper establishes a natural -multicategory structure on moduli spaces of pseudo-holomorphic polygons to provide a uniform operadic framework for various -type structures, including Fukaya categories, as algebras over differential graded -multicategories.
This paper establishes that dynamically convex, closed contact type hypersurfaces in enclosing the zero section possess at least closed Reeb orbits, and further guarantees the existence of at least two irrationally elliptic orbits under non-degeneracy and finiteness conditions.
The paper establishes that topological symplectic manifolds possess a canonical bi-Lipschitz structure, a result that yields the first known examples demonstrating both the non-existence and non-uniqueness of topological symplectic structures.
This paper introduces a novel construction of local Morse homology to compute the homology of critical points in the Lagrange problem, thereby proving that linear critical points are either saddle points or degenerate, refining the previous conclusion that they must be saddle points if non-degenerate.
This paper presents a symplectic Hamiltonian framework for thermodynamics that models equilibrium states as Lagrangian submanifolds and describes both reversible and irreversible processes, such as free expansion and heat transfer, through Hamiltonian dynamics and port-Hamiltonian systems.
This paper establishes a framework of Courant algebroid relations to connect Dirac structures, spinors, and generalised complex/Kähler geometries, demonstrating how these relations induce T-duality in supersymmetric sigma-models and remain compatible with Type II supergravity equations.
This paper introduces "Hodge atoms," new birational invariants constructed by combining rational Gromov-Witten invariants with Hodge theory via F-bundles, which are used to prove the irrationality of very general cubic fourfolds, reprove the equality of Hodge numbers for birational Calabi-Yau manifolds, and provide new obstructions to rationality over non-algebraically closed fields.
This paper proves that for any reversible Finsler metric on the two-sphere, the number of prime closed geodesics grows quadratically with respect to length, utilizing improved versions of Franks' theorem and cylindrical contact homology.
This paper introduces a new class of Liouville polarizations for open symplectic manifolds to establish symplectic embedding results, demonstrate non-removable Lagrangian intersections at small scales, and reveal a novel phenomenon of Legendrian barriers in contact geometry.
This paper establishes a unified framework using symplectic field theory functorial properties to generate finite algebraic (planar) torsions, thereby confirming a conjecture by Latschev and Wendl regarding stably fillable examples in dimensions five and higher and demonstrating the ubiquity of tight, non-weakly fillable contact structures on spheres in those dimensions.
The paper proves that any odd-dimensional parallelisable closed manifold admits a contact structure, thereby establishing that odd-dimensional solvmanifolds are contact.
The paper demonstrates the existence of non-formal compact (almost) contact manifolds with specific first Betti numbers and, in certain dimensions, simple connectivity, thereby resolving a key problem in the geography of such manifolds.
This paper provides a complete symplectic-topological classification of periodic orbits in the spatial rotating Kepler problem by introducing a new coordinate system based on the Laplace-Runge-Lenz vector, computing their Conley-Zehnder and Robbin-Salamon indices, and determining their contributions to symplectic homology via the Morse-Bott spectral sequence.