We show that a nondegenerate tight contact form on the 3-sphere has exactly two simple closed Reeb orbits if and only if the differential in linearized contact homology vanishes. Moreover, in this case the Floquet multipliers and Conley-Zehnder indices of the two Reeb orbits agree with those of a suitable irrational ellipsoid in 4-space.
We study the combinatorial Reeb flow on the boundary of a four-dimensional convex polytope. We establish a correspondence between combinatorial Reeb orbits for a polytope, and ordinary Reeb orbits for a smoothing of the polytope, respecting action an
d Conley-Zehnder index. One can then use a computer to find all combinatorial Reeb orbits up to a given action and Conley-Zehnder index. We present some results of experiments testing Viterbos conjecture and related conjectures. In particular, we have found some new examples of polytopes with systolic ratio $1$.
We generalize Bangerts non-hyperbolicity result for uniformly tamed almost complex structures on standard symplectic $R^{2n}$ to asymtotically standard symplectic manifolds.
We study tight projective 2-designs in three different settings. In the complex setting, Zauners conjecture predicts the existence of a tight projective 2-design in every dimension. Pandey, Paulsen, Prakash, and Rahaman recently proposed an approach
to make quantitative progress on this conjecture in terms of the entanglement breaking rank of a certain quantum channel. We show that this quantity is equal to the size of the smallest weighted projective 2-design. Next, in the finite field setting, we introduce a notion of projective 2-designs, we characterize when such projective 2-designs are tight, and we provide a construction of such objects. Finally, in the quaternionic setting, we show that every tight projective 2-design for H^d determines an equi-isoclinic tight fusion frame of d(2d-1) subspaces of R^d(2d+1) of dimension 3.
In this short note we observe that the boundary of a properly embedded compact exact Lagrangian sub-manifolds in a subcritical Weinstein domain $X$ necessarily admits Reeb chords. The existence of this Reeb chords either follows from an obstruction t
o the deformation of the boundary to a cylinder over a Legendrian sub-manifold or from the fact that the wrapped Floer homology of the Lagrangian vanishes once this boundary have been collared.
We show that the Hamiltonian action satisfies the Palais-Smale condition over a mixed regularity space of loops in cotangent bundles, namely the space of loops with regularity $H^s$, $sin (frac 12, 1)$, in the base and $H^{1-s}$ in the fiber directio
n. As an application, we give a simplified proof of a theorem of Hofer-Viterbo on the existence of closed characteristic leaves for certain contact type hypersufaces in cotangent bundles.