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Symplectic flux measures the areas of cylinders swept in the process of a Lagrangian isotopy. We study flux via a numerical invariant of a Lagrangian submanifold that we define using its Fukaya algebra. The main geometric feature of the invariant is its concavity over isotopies with linear flux. We derive constraints on flux, Weinstein neighbourhood embeddings and holomorphic disk potentials for Gelfand–Cetlin fibres of Fano varieties in terms of their polytopes. We also describe the space of fibres of almost toric fibrations on the complex projective plane up to Hamiltonian isotopy, and provide other applications. [ABSTRACT FROM AUTHOR]

We study configurations of disjoint Lagrangian submanifolds in certain low-dimensional symplectic manifolds from the perspective of the geometry of Hamiltonian maps. We detect infinite-dimensional flats in the Hamiltonian group of the two-sphere equipped with Hofer's metric, prove constraints on Lagrangian packing, find instances of Lagrangian Poincaré recurrence, and present a new hierarchy of normal subgroups of area-preserving homeomorphisms of the two-sphere. The technology involves Lagrangian spectral invariants with Hamiltonian term in symmetric product orbifolds. [ABSTRACT FROM AUTHOR]

We identify a new class of closed smooth manifolds for which there exists a uniform bound on the Lagrangian spectral norm of Hamiltonian deformations of the zero section in a unit cotangent disk bundle. This settles a well-known conjecture of Viterbo from 2007 as the special case of T n , which has been completely open for n > 1 . Our methods are different and more intrinsic than those of the previous work of the author first settling the case n = 1 . The new class of manifolds is defined in topological terms involving the Chas–Sullivan algebra and the BV-operator on the homology of the free loop space. It contains spheres and is closed under products. We discuss generalizations and various applications, to C 0 symplectic topology in particular. [ABSTRACT FROM AUTHOR]

We prove a conjecture of Viterbo from 2007 on the existence of a uniform bound on the Lagrangian spectral norm of Hamiltonian deformations of the zero section in unit cotangent disk bundles, for bases given by compact rank one symmetric spaces S n , R P n , C P n , H P n , n ≥ 1. We discuss generalizations and give applications, in particular to C 0 symplectic topology. Our key method consists in a quantitative deformation argument for Floer persistence modules that allows to excise a divisor. [ABSTRACT FROM AUTHOR]

We prove a number of new results on the large-scale geometry of the L p -metrics on the group of area-preserving diffeomorphisms of each orientable surface. Our proofs use in a key way the Fulton-MacPherson type compactification of the configuration space of n points on the surface due to Axelrod-Singer and Kontsevich. This allows us to apply the Schwarz-Milnor lemma to configuration spaces, a natural approach which we carry out successfully for the first time. As sample results, we prove that all right-angled Artin groups admit quasi-isometric embeddings into the group of area-preserving diffeomorphisms endowed with the L p -metric, and that all Gambaudo-Ghys quasi-morphisms on this metric group coming from the braid group on n strands are Lipschitz. This was conjectured to hold, yet proven only for small values of n and g, where g is the genus of the surface. [ABSTRACT FROM AUTHOR]

In this note we prove that if a closed monotone symplectic manifold M of dimension 2n, satisfying a homological condition that holds in particular when the minimal Chern number is N > n, admits a Hamiltonian pseudo-rotation, then the quantum Steenrod square of the point class must be deformed. This gives restrictions on the existence of pseudo-rotations. Our methods rest on previous work of the author, Zhao, and Wilkins, going back to the equivariant pair-of-pants product-isomorphism of Seidel. [ABSTRACT FROM AUTHOR]

We prove that the autonomous norm on the group of Hamiltonian diffeomorphisms of the two-dimensional torus is unbounded. We provide examples of Hamiltonian diffeomorphisms with arbitrarily large autonomous norm. For the proofs we construct explicit quasimorphisms on some of them are -continuous and vanish on all autonomous diffeomorphisms, and some of them are Calabi. [ABSTRACT FROM AUTHOR]

We find robust obstructions to representing a Hamiltonian diffeomorphism as a full k-th power, $$k \ge 2$$ , and in particular, to including it into a one-parameter subgroup. The robustness is understood in the sense of Hofer's metric. Our approach is based on the theory of persistence modules applied in the context of filtered Floer homology. We present applications to geometry and dynamics of Hamiltonian diffeomorphisms. [ABSTRACT FROM AUTHOR]

In this paper, we prove the main conjecture on g-areas arising from [7]. That conjecture was announced by the first author in 2004. It states that the g-area of any Hamiltonian diffeomorphism ø is equal to the positive Hofer pseudo-distance between ø and the subspace of Hamiltonian diffeomorphisms that can be expressed as a product of at most g commutators. [ABSTRACT FROM AUTHOR]

We establish a C0 a priori bound on the solutions of the quaternionic Calabi-Yau equation (of Monge-Ampère type) on compact HKT manifolds with a locally flat hypercomplex structure. As an intermediate step, we prove a quaternionic version of the Gauduchon theorem. [ABSTRACT FROM AUTHOR]