# Symplectic Geometry

## New submissions

[ total of 6 entries: 1-6 ]
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### New submissions for Tue, 20 Mar 18

[1]
Title: Global surfaces of section for dynamically convex Reeb flows on lens spaces
Subjects: Symplectic Geometry (math.SG); Dynamical Systems (math.DS)

We show that a dynamically convex Reeb flow on a lens space $L(p, 1), p>1,$ admits a $p$-unknotted closed Reeb orbit $P$ which is the binding of a rational open book decomposition with disk-like pages. Each page is a rational global surface of section for the Reeb flow and the Conley-Zehnder index of the $p$-th iterate of $P$ is $3$. This result applies to the H\'enon-Heiles Hamiltonian whose energy level presents $\mathbb{Z}_3$-symmetric and for all energies $<1/6$ the flow restricted to the sphere-like component descends to a dynamically convex Reeb flow on $L(3,1)$. Due to a $\mathbb{Z}_4$-symmetry the result also applies to Hill's lunar problem.

[2]
Title: The decomposition formula for Verlinde Sums
Subjects: Symplectic Geometry (math.SG)

We prove a decomposition formula for Verlinde sums (rational trigonometric sums), as a discrete counterpart to the Boysal-Vergne decomposition formula for Bernoulli series. Motivated by applications to fixed point formulas in Hamiltonian geometry, we develop differential form valued version of Bernoulli series and Verlinde sums, and extend the decomposition formula to this wider context.

### Cross-lists for Tue, 20 Mar 18

[3]  arXiv:1803.06455 (cross-list from math.GT) [pdf, ps, other]
Title: A-infinity algebras, strand algebras, and contact categories
Comments: 83 pages, 23 figures, 6 tables
Subjects: Geometric Topology (math.GT); Algebraic Topology (math.AT); Symplectic Geometry (math.SG)

In previous work we showed that the contact category algebra of a quadrangulated surface is isomorphic to the homology of a strand algebra from bordered Floer theory. Being isomorphic to the homology of a differential graded algebra, this contact category algebra has an A-infinity structure. In this paper we investigate such A-infinity structures in detail. We give explicit constructions of such A-infinity structures, and establish some of their properties, including conditions for the nonvanishing of A-infinity operations. Along the way we develop several related notions, including a detailed consideration of tensor products of strand diagrams.

[4]  arXiv:1803.06970 (cross-list from math.DG) [pdf, ps, other]
Title: Higher symmetries of symplectic Dirac operator
Comments: Symplectic Dirac operator, Higher symmetry algebra, Projective differential geometry, Minimal nilpotent orbit, $\mathfrak{sl}(3,\mR)$
Subjects: Differential Geometry (math.DG); Mathematical Physics (math-ph); Functional Analysis (math.FA); Representation Theory (math.RT); Symplectic Geometry (math.SG)

We construct in projective differential geometry of the real dimension $2$ higher symmetry algebra of the symplectic Dirac operator ${D}\kern-0.5em\raise0.22ex\hbox{/}_s$ acting on symplectic spinors. The higher symmetry differential operators correspond to the solution space of a class of projectively invariant overdetermined operators of arbitrarily high order acting on symmetric tensors. The higher symmetry algebra structure corresponds to a completely prime primitive ideal having as its associated variety the minimal nilpotent orbit of $\mathfrak{sl}(3,{\mathbb{R}})$.

### Replacements for Tue, 20 Mar 18

[5]  arXiv:1707.00977 (replaced) [pdf, ps, other]
Title: Electric-Magnetic Aspects On Yang-Mills Fields
Authors: Tosiaki Kori