Symplectically degenerate maxima via generating functions

We provide a simple proof of a theorem due to Nancy Hingston, asserting that symplectically degenerate maxima of any Hamiltonian diffeomorphism of the standard symplectic 2d-torus are non-isolated contractible periodic points or their action is a non-isolated point of the average-action spectrum. Our argument is based on generating functions.Comment: 25 pages, thoroughly revised version, new titl

The Conley Conjecture and Beyond

This is (mainly) a survey of recent results on the problem of the existence of infinitely many periodic orbits for Hamiltonian diffeomorphisms and Reeb flows. We focus on the Conley conjecture, proved for a broad class of closed symplectic manifolds, asserting that under some natural conditions on the manifold every Hamiltonian diffeomorphism has infinitely many (simple) periodic orbits. We discuss in detail the established cases of the conjecture and related results including an analog of the conjecture for Reeb flows, the cases where the conjecture is known to fail, the question of the generic existence of infinitely many periodic orbits, and local geometrical conditions that force the existence of infinitely many periodic orbits. We also show how a recently established variant of the Conley conjecture for Reeb flows can be applied to prove the existence of infinitely many periodic orbits of a low-energy charge in a non-vanishing magnetic field on a surface other than a sphere.Comment: 34 pages, 1 figur

PORTEC-4a: International randomized trial of molecular profile-based adjuvant treatment for women with high-intermediate risk endometrial cancer

Background Vaginal brachytherapy is currently recommended as adjuvant treatment in patients with highintermediate risk endometrial cancer to maximize local control and has only mild side effects and no or limited impact on quality of life. However, there is still considerable overtreatment and also some undertreatment, which may be reduced by tailoring adjuvant treatment to the patients’ risk of recurrence based on molecular tumor characteristics. Primary objectives To compare the rates of vaginal recurrence in women with high-intermediate risk endometrial cancer, treated after surgery with molecularintegrated risk profile-based recommendations for either observation, vaginal brachytherapy or external pelvic beam radiotherapy or with standard adjuvant vaginal brachytherapy Study hypothesis Adjuvant treatment based on a molecular-integrated risk profile provides similar local control and recurrence-free survival as current standard adjuvant brachytherapy in patients with high-intermediate risk endometrial cancer, while sparing many patients the morbidity of adjuvant t