European navigation in the age of sail owes much to the scientific revolution of the seventeenth century and the development of instruments and advanced mathematical techniques. Important though these developments were, it is argued here that close observation of the environment: of the weather, ocean currents, clouds, birds, mammals, and a host of other factors played a far more important role in safe navigation from one part of the globe to another.

DeMarco, Krieger, and Ye conjectured that there is a uniform bound B, depending only on the degree d, so that any pair of holomorphic maps $f, g :{\mathbb {P}}^1\to {\mathbb {P}}^1$ with degree $d$ will either share all of their preperiodic points or have at most $B$ in common. Here we show that this uniform bound holds for a Zariski open and dense set in the space of all pairs, $\mathrm {Rat}_d \times \mathrm {Rat}_d$, for each degree $d\geq 2$. The proof involves a combination of arithmetic intersection theory and complex-dynamical results, especially as developed recently by Gauthier and Vigny, Yuan and Zhang, and Mavraki and Schmidt. In addition, we present alternate proofs of the main results of DeMarco, Krieger, and Ye [Uniform Manin-Mumford for a family of genus 2 curves, Ann. of Math. (2) 191 (2020), 949–1001; Common preperiodic points for quadratic polynomials, J. Mod. Dyn. 18 (2022), 363–413] and of Poineau [Dynamique analytique sur $\mathbb {Z}$ II : Écart uniforme entre Lattès et conjecture de Bogomolov-Fu-Tschinkel, Preprint (2022), arXiv:2207.01574 [math.NT]]. In fact, we prove a generalization of a conjecture of Bogomolov, Fu, and Tschinkel in a mixed setting of dynamical systems and elliptic curves.

We consider the total curvature of graphs of curves in high-codimension Euclidean space. We introduce the corresponding relaxed energy functional and prove an explicit representation formula. In the case of continuous Cartesian curves, i.e. of graphs cu of continuous functions u on an interval, we show that the relaxed energy is finite if and only if the curve cu has bounded variation and finite total curvature. In this case, moreover, the total curvature does not depend on the Cantor part of the derivative of u. We treat the wider class of graphs of one-dimensional functions of bounded variation, and we prove that the relaxed energy is given by the sum of the length and total curvature of the new curve obtained by closing the holes in cu generated by jumps of u with vertical segments.

The circulation and water masses of Port Foster, Deception Island, were studied using conductivity-temperature-depth stations inside and outside the semi-enclosed bay and an array of bottom temperature sensors moored around the perimeter of the bay over two weeks in the summer of 2012. Inside Port Foster, the water column is divided into two layers separated by a temperature-forced, seasonal pycnocline at ~40–60 m. The circulation of the upper layer is in an anticlockwise direction, with mean geostrophic currents of ~0.04–0.10 ms-1. The lower layer, from ~60 m to the seabed, shows coastal-trapped waves travelling in a clockwise direction, possibly triggered by local wind gusts. Local sea ice melt in areas surrounding the underwater hot springs of Pendulum Cove appears as a fresh, warm anomaly down to 30 m.

We prove existence of minimizing movements for thedislocation dynamics evolution law of a propagating front, in which the normal velocity of the front is the sum of a non-local term and a mean curvature term. We prove that any such minimizing movement is a weak solution of this evolution law, in a sense related to viscositysolutions of the corresponding level-set equation. We also prove theconsistency of this approach, by showing that any minimizing movementcoincides with the smooth evolution as long as the latter exists. Inrelation with this, we finally prove short time existence and uniqueness of a smoothfront evolving according to our law, provided the initial shape issmooth enough.

Current measurements obtained from a sparse array of moorings on the Ross Continental Shelf during the 1980 and 90s are analysed for their tidal constituents. Diurnal (K1 and O1) tides are about five times stronger than semi-diurnal tides, and are strongest near the shelf break in agreement with recent model results. At some energetic locations the diurnal tides are significantly weaker at depth, presumably due to bottom friction. Complex demodulation analysis shows that at some locations the tidal response varies significantly in time. This time variability rises markedly above the noise floor expected from the spectral continuum between tidal bands, and does not scale with tidal bandwidth as would be expected of broadband noise. Further its magnitude is generally proportional to the associated tidal constituent, indicating that it is truly a varying tidal response. Space scales of this tidal instability were not resolved but are less than 150 km, while time scales appear mostly seasonal to interannual. The rms magnitude of the unstable response can be 1/3 of a given component’s mean magnitude, placing substantial limits on the ability of prognostic or even data assimilative models to accurately predict these tides for any specific time period.