An integer valued bi-invariant metric on the group of contactomorphisms of R^2n x S^1

In his 1992 article on generating functions Viterbo constructed a bi-invariant metric on the group of compactly supported Hamiltonian symplectomorphisms of R^2n. Using the set-up of arXiv:0901.3112 we extend the Viterbo metric to the group of compactly supported contactomorphisms of R^2n x S^1 isotopic to the identity. We also prove that the contactomorphism group is unbounded with respect to this metric.Comment: 10 pages, final versio

Bi-invariant metrics on contactomorphism groups

Contact manifolds are odd-dimensional smooth manifolds endowed with a maximally non-integrable field of hyperplanes. They are intimately related to symplectic manifolds, i.e. even-dimensional smooth manifolds endowed with a closed non-degenerate 2-form. Although in symplectic topology a famous bi-invariant metric, the Hofer metric, has been studied since more than 20 years ago, it is only recently that some somehow analogous bi-invariant metrics have been discovered on the group of diffeomorphisms that preserve a contact structure. In this expository article I will review some constructions of bi-invariant metrics on the contactomorphism group, and how these metrics are related to some other global rigidity phenomena in contact topology which have been discovered in the last few years, in particular the notion of orderability (due to Eliashberg and Polterovich) and an analogue in contact topology (due to Eliashberg, Kim and Poltorovich) of Gromov's symplectic non-squeezing theorem.Comment: Expository article, written in occasion of the 5-th IST-IME meeting in S\~ao Paulo (July 2014) in honor of Prof. Orlando Lopes. To appear in the S\~ao Paulo Journal of Mathematical Science

Invoice from Sandon & Co. to Robert Goelet

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Anterior cruciate ligament injuries in football

Anterior cruciate ligament (ACL) injuries and football are clearly intertwined. ACL injuries are principally sustained when participating in pivoting activities and, as it is such a popular sport, football is consequently a major cause of these injuries. The flip side of this is of course that ACL injuries negatively affect the afflicted players’ potential to play the game. It is this relationship the present thesis attempts to elucidate. In Study I, factors associated with a return to football after ACL reconstruction were examined in football players. The participants were asked about their return to football and knee symptoms during physical activity. Data were collected from Capio Artro Clinics local database regarding surgical information, patient-reported outcome measurements (PROMs), knee laxity and muscle strength. In the study, female gender, cartilage injury and knee pain during physical activity were found to be independent negative factors for returning to football after ACL reconstruction. In Study II, a 10-year follow-up of football players from all levels in the Swedish National Knee Ligament Registry (SNKLR) was performed using a questionnaire to investigate how many players returned to play, what influenced their decision and whether there were any differences in additional ACL injuries (graft failure and/or contralateral ACL injury) between those who returned to play and those who did not. Fifty-four per cent of the football players returned to play after ACL reconstruction and, in two-thirds of those who did not return, the reason was knee related. Players who returned to football ran a significantly higher risk of sustaining further ACL injury. In Study III, the effect of an ACL reconstruction for teenage football talents on their potential to continue as football players and become elite players at senior level was investigated. All players from the NEC (national elite camps) over six seasons were followed and data were collected from the SNKLR and the Swedish FA. As no significant differences were found between the ACL-reconstructed and uninjured players, it appears that ACL-reconstructive surgery in talented youth football players offers them the opportunity to become elite players as seniors and permits an activity level on a par with that of their uninjured peers. In Study IV, an epidemiological survey was made of players who, on the website of the SNKLR, provided information on football-specific factors associated with their ACL injury. The study found that ACL injuries were more frequently sustained during games, 66%, than during practice, 25%. Knee control exercises to warm up were used by 31% of the female players and 16% of the males. Forty per cent of the players reported that they did not plan on returning to football. Teams changing coaches and players moving to a higher division appear to run an increased risk of ACL injury

Givental's non-linear Maslov index on lens spaces

Givental's non-linear Maslov index, constructed in 1990, is a quasimorphism on the universal cover of the identity component of the contactomorphism group of real projective space. This invariant was used by several authors to prove contact rigidity phenomena such as orderability, unboundedness of the discriminant and oscillation metrics, and a contact geometric version of the Arnold conjecture. In this article we give an analogue for lens spaces of Givental's construction and its applications.Comment: 44 pages; v3: minor changes; v2: besides minor changes, we corrected a mistake in Corollary 1.3(iv