The symplectic Floer homology HF_*(f) of a symplectomorphism f:S->S encodes data about the fixed points of f using counts of holomorphic cylinders in R x M_f, where M_f is the mapping torus of f. We give an algorithm to compute HF_*(f) for f a surface symplectomorphism in a pseudo-Anosov or reducible mapping class, completing the computation of Seidel's HF_*(h) for h any orientation-preserving mapping class.Comment: 57 pages, 4 figures. Revision for publication, with various minor corrections. Adds results on the module structure and invariance thereo

Given a closed, oriented surface Σ, possibly with boundary, and a mapping class, we obtain sharp lower bounds on the number of fixed points of a surface symplectomorphism (i.e. area-preserving map) in the given mapping class, both with and without nondegeneracy assumptions on the fixed points. This generalizes the Poincaré-Birkhoff fixed point theorem to arbitrary surfaces and mapping classes. These bounds often exceed those for non-areapreserving maps. We obtain these bounds from Floer homology computations with certain twisted coefficients plus a method for obtaining fixed point bounds on entire symplectic mapping classes on monotone symplectic manifolds from such computations. For the case of possibly degenerate fixed points, we use quantum-cup-length-type arguments for certain cohomology operations we define on summands of the Floer homology.1 Our results extend to the nonnegative Euler characteristic case but these exceptional cases would be cumbersome to carry around. All of these cases are already understood.2 That is, the fixed points of φ are cut out transversally in the sense that det(1 − dφx) = 0 for fixed points x.

Introduction American adults are typically sleep deprived during weekdays and attempt to recover sleep on the weekends. Technological advances in home sleep monitoring have provided the opportunity to analyze sleep patterns on a scale much larger than previously imaginable. This study explores the weekly REM sleep deprivation–recovery cycle in a large U.S. sample. Methods Estimated total sleep time (TST) and REM/TST (R%) were analyzed by a commercially-available home-sleep-monitoring device (Sleeptracker-AI Monitor, Fullpower Technologies, California, USA). The device passively monitors sleep using piezo-electric sensors that register the forces exerted through the mattress. The de-identified data from the devices were analyzed following review and exemption of the study (#57681) from the Stanford University IRB. Data from 07/2020–06/2021, from 101,442 individuals with 14,277,964 recorded nights, were available. The analytic dataset included individuals with at least 300 nights of sleep per year and 26 of 52 nights per each day of the week (excluding nights abutting federal holidays). Results A total of 21,543 individuals (11,095 men, 51±14 years; 9,821 women, 50±15 years; 627 unspecified genders) and 6,850,717 recorded nights met the inclusion criteria. There is a stepwise increase in R% from Sunday night to Friday night and a decrease back to Sunday night, following a cycle of weekday sleep deprivation and weekend recovery. The means and standard deviations (across individuals’ averages) of TST in hours and R% for each night were: Sunday (TST*:7.21±0.885, R%*:24.20±3.09), Monday (TST*:7.18±0.853, R%*:24.56±3.10), Tuesday (TST*:7.16±0.847, R%*:24.67±3.13), Wednesday (TST*:7.16±0.845, R%*:24.80±3.15), Thursday (TST*:7.18±0.845, R%*:24.87±3.15), Friday (TST*:7.51±0.904, R%*:25.05±3.15), and Saturday (TST*:7.59±0.897, R%*:24.83±3.12). Each statistic, when compared with the previous night of the week, was significant (p < 0.05/7, Bonferroni corrected) by paired t-test (denoted by an asterisk). Conclusion The use of advanced technology to estimate sleep-wake patterns in a large sample permits the validation of a repetitive REM sleep deprivation–recovery cycle. Individuals are, on average, partially sleep deprived starting Sunday night, which leads to a progressive REM sleep rebound that transitions into a REM recovery cycle on Friday and Saturday nights. Further work will focus on studying this cycle within different groups (e.g., age, gender), across seasons, and including other sleep parameters. Support (If Any)

A sharp lower bound on fixed points of surface symplectomorphisms in each mapping class ANDREW COTTON-CLAY msp

Introduction Sharing the bed with a partner is common among adults and is likely to impact sleep in multiple ways. However, polysomnograms are performed without a bed partner and objective data on co-sleeping couples are extremely rare. This study aimed to investigate the effects of a bed partner's presence on objective sleep parameters. Methods Sleep data from 5190 users (43% female, 14% unspecified gender, mean age 47) and their bed partners were collected through a commercially-available home sleep monitoring device (Sleeptracker-AI Monitor, Fullpower Technologies, California, USA). The device passively monitors sleep using piezo-electric sensors that register the forces exerted through the mattress. Subjects with at least 10 weekday sleep recordings with a bed partner present for at least an hour, and at least 10 weekday sleep recordings without a bed partner present at all during the period from 08/2021 to 11/2021 were included. Estimated total sleep time (TST), wake after sleep onset (WASO), sleep efficiency (SE), and light, deep, and REM sleep were analyzed comparing between the nights with and without a bed partner. Results The mean (standard deviation) across subject averages of estimated TST (min), WASO (min), SE (%), and light, deep and REM sleep (min) with a bed partner were: TST 417.2(44.5), WASO 47.2(24.9), SE 89.5(6.4), light sleep 255.4(34.2), deep sleep 56.7(11.9) and REM sleep 105.2(17.5) and without a bed partner were: TST 414.7(49.2)-*, WASO 40.7(20.5)-*, SE 90.7(5.8)+*, light sleep 243.6(38.0)-*, deep sleep 62.9(12.9)+* and REM sleep 108.2(19.0)+*; a (+) indicates an increase and (-) a decrease in the sleep parameter between nights with and without a bed partner, and (*) signifies p < 0.05 by paired t-test. Conclusion When the bed partner is absent, an individual’s sleep architecture shows on average a higher sleep efficiency, with less awake time but also less total sleep; more minutes spent in deep and REM sleep, and less in light sleep. This suggests a less interrupted night, perhaps due to fewer disruptions from the partner, where the individual has enough continuity in his/her sleep to transition to deeper stages. Further work will add the effect of a bedpartner in AHI and snoring. Support (If Any)