We study two related invariants of Lagrangian submanifolds in symplectic manifolds. For a Lagrangian torus these invariants are functions on the first cohomology of the torus.The first invariant is of topological nature and is related to the study of Lagrangian isotopies with a given Lagrangian flux. More specifically, it measures the length of straight paths in the first cohomology that can be realized as the Lagrangian flux of a Lagrangian isotopy.The second invariant is of analytical nature and comes from symplectic function theory. It is defined for Lagrangian submanifolds admitting fibrations over a circle and has a dynamical interpretation.We partially compute these invariants for certain Lagrangian tori.
We suggest a homotopical description of the Poisson bracket invariants for tuples of closed sets in symplectic manifolds. It implies that these invariants depend only on the union of the sets along with topological data. 1 Introduction 995 2 ε-pseudoretracts 1000 3 The invariants Pb n 1003 4 The invariant Pb X (α) 1017 References 1026We define invariants, Pb N , of N -tuples of cyclically intersecting compact subsets of a symplectic manifold M . For a compact M they can be defined
We enumerate complex curves on toric surfaces of any given degree and genus, having a single cusp and nodes as their singularities, and matching appropriately many point constraints. The solution is obtained via tropical enumerative geometry. The same technique applies to enumeration of real plane cuspidal curves: we show that, for any fixed $r\ge 1$ and $d\ge 2r+3$, there exists a generic real $2r$-dimensional linear family of plane curves of degree $d$ in which the number of real $r$-cuspidal curves is asymptotically comparable with the total number of complex $r$-cuspidal curves in the family, as $d\to \infty $.
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