Global surfaces of section for Reeb flows in dimension three and beyond

We survey some recent developments in the quest for global surfaces of section for Reeb flows in dimension three using methods from Symplectic Topology. We focus on applications to geometry, including existence of closed geodesics and sharp systolic inequalities. Applications to topology and celestial mechanics are also presented.Comment: 33 pages, 3 figures. This is an extended version of a paper written for Proceedings of the ICM, Rio 2018; in v3 we made minor additional corrections, updated references, added a reference to work of Lu on the Conley Conjectur

Conditioning temperature for inducing uniform ripening of 'Abate Fetel' pears.

The aim of this study was to determine the conditioning temperature period for inducing uniform ripening of ‘Abate Fetel’ pears as well as to determine the maximum storage period with no quality loss. Three harvests were carried out at weekly intervals and each harvest date was considered a factorial experiment (7x3), with seven storage periods at 0 ± 1 °C and 90 ± 5% relative humidity (0; 20; 40; 60; 80; 100 and 120 days) and three periods at room temperature (20 ± 1 °C) (zero, three and six days). The assessed quality attributes were flesh firmness, starch content, weight loss and soluble solids content. Acceptance and purchase intention sensory tests were carried out. Cold storage was efficient in inducing uniform ripening in ‘Abate Fetel’ pears after 20 days at low temperature followed by six days at room temperature for fruit from the three harvest dates. ‘Abate Fetel’ pears harvested at harvest maturity of 57 N can be stored for up to 120 days and commercialized within six days with no quality loss. However, fruit harvested at flesh firmness below 55 N can be stored for 80 days and commercialized within six days or stored for 100 days and commercialized within three days. Key words: Pyrus communis L.. Cold storage. Acceptance. Shelf life. O objetivo deste trabalho foi de definir o tempo necessário de condicionamento por frio para causar a uniformização do amadurecimento de peras ‘Abate Fetel’. Ainda, objetivou-se a determinação do período máximo de armazenamento sem que houvesse prejuízos à qualidade de frutos. Foram realizadas três colheitas em intervalos semanais e cada colheita constituiu um experimento fatorial (7x3), sendo sete períodos de manutenção sob baixa temperatura (0; 20; 40; 60; 80; 100 e 120 dias) e três períodos de manutenção em ambiente simulado a 20 ± 1 °C (zero, três e seis dias). Os atributos de qualidade avaliados foram firmeza de polpa, índice de degradação do amido, perda de massa fresca, teores de sólidos solúveis e aceitabilidade. O condicionamento por frio foi eficiente na indução e uniformização do amadurecimento de peras ‘Abate Fetel’, após 20 dias de manutenção em baixa temperatura seguidos de seis dias em ambiente simulado, para os frutos provenientes das três datas de colheita. Frutos da cultivar Abate Fetel com data de colheita DC1 podem ser armazenados por até 120 dias e comercializados em até seis dias sem que haja comprometimento da qualidade. Já frutos colhidos nas datas de colheita DC2 e DC3 podem ser armazenados por 80 dias e comercializados em até seis dias ou armazenados por 100 dias e comercializados no período de três dias. Palavras-chave: Pyrus communis L.. Armazenamento refrigerado. Aceitabilidade

Global properties of tight Reeb flows with applications to Finsler geodesic flows on S2S^2

We show that if a Finsler metric on $S^2$ with reversibility $r$ has flag curvatures $K$ satisfying $(\frac{r}{r+1})^2 <K \leq 1$, then closed geodesics with specific contact-topological properties cannot exist, in particular there are no closed geodesics with precisely one transverse self-intersection point. This is a special case of a more general phenomenon, and other closed geodesics with many self-intersections are also excluded. We provide examples of Randers type, obtained by suitably modifying the metrics constructed by Katok \cite{katok}, proving that this pinching condition is sharp. Our methods are borrowed from the theory of pseudo-holomorphic curves in symplectizations. Finally, we study global dynamical aspects of 3-dimensional energy levels $C^2$-close to $S^3$.Comment: 27 page