Ana Rechtman

THE WEINSTEIN CONJECTURE IN THE PRESENCE OF SUBMANIFOLDS HAVING A LEGENDRIAN FOLIATION

Helmut Hofer introduced in 1993 a novel technique based on holomorphic curves to prove the Weinstein conjecture. Among the cases where these methods apply are all contact 3-manifolds (M, ξ) with π2(M) ≠ 0. We modify Hofers argument to prove the Weinstein conjecture for some examples of higher-dimensional contact manifolds. In particular, we are able to show that the connected sum with a real projective space always has a closed contractible Reeb orbit.

Existence of periodic orbits for geodesible vector fields on closed 3-manifolds

In this paper we deal with the existence of periodic orbits of geodesible vector fields on closed 3-manifolds. A vector field is geodesible if there exists a Riemannian metric on the ambient manifold making its orbits geodesics. In particular, Reeb vector fields and vector fields that admit a global section are geodesible. We will classify the closed 3-manifolds that admit aperiodic volume preserving real analytic geodesible vector fields, and prove the existence of periodic orbits for real analytic geodesible vector fields (not volume preserving), when the 3-manifold is not a torus bundle over the circle. We will also prove the existence of periodic orbits of C2 geodesible vector fields in some closed 3-manifolds.

Equivalence of deterministic walks on regular lattices on the plane

We consider deterministic walks on square, triangular and hexagonal two dimensional lattices. In each case, there is a scatterer at every lattice site that can be in one of two states that forces the walker to turn either to his/her immediate right or left. After the walker is scattered, the scatterer changes state. A lattice with an arrangement of scatterers is an environment. We show that there are only two environments for which the scattering rules are injective, mirrors or rotators, on the three lattices. On hexagonal lattices Webb and Cohen (2014), proved that if a walker with a given initial position and velocity moves through an environment of mirrors (rotators) then there is an environment of rotators (mirrors) through which the walker would move with the same trajectory. We refer to these trajectories on mirror and rotator environments as equivalent walks. We prove the equivalence of walks on square and triangular lattices and include a proof of the equivalence of walks on hexagonal lattices. The proofs are based both on the geometry of the lattice and the structure of the scattering rule.