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1. Constructions of Lagrangian cobordisms

   https://doi.org/10.1007/978-3-030-80979-9  

   Blackwell, Sarah; Legout, Noémie; Leverson, Caitlin; Limouzineau, Maÿlis; Myer, Ziva; Pan, Yu; Pezzimenti, Samantha; Simone Suárez, Lara; Traynor, Lisa (January 2021, Association for Women in Mathematics series)

   Acu, Bahar; Cannizzo, Catherine; McDuff, Dusa; Myer, Ziva; Pan, Yu; Traynor, Lisa (Ed.)

   Lagrangian cobordisms between Legendrian knots arise in Symplectic Field Theory and impose an interesting and not well-understood relation on Legendrian knots. There are some known ``elementary'' building blocks for Lagrangian cobordisms that are smoothly the attachment of 0- and 1-handles. An important question is whether every pair of non-empty Legendrians that are related by a connected Lagrangian cobordism can be related by a ribbon Lagrangian cobordism, in particular one that is ``decomposable'' into a composition of these elementary building blocks. We will describe these and other combinatorial building blocks as well as some geometric methods, involving the theory of satellites, to construct Lagrangian cobordisms. We will then survey some known results, derived through Heegaard Floer Homology and contact surgery, that may provide a pathway to proving the existence of nondecomposable (nonribbon) Lagrangian cobordisms. 

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2. A Global Comparison of Marine Chlorophyll Variability Observed in Eulerian and Lagrangian Perspectives

   https://doi.org/10.1029/2023JC019801  

   Kuhn, Angela M.; Mazloff, Matthew; Dutkiewicz, Stephanie; Jahn, Oliver; Clayton, Sophie; Rynearson, Tatiana; Barton, Andrew D. (July 2023, Journal of Geophysical Research: Oceans)

   Abstract Ocean chlorophyll time series exhibit temporal variability on a range of timescales due to environmental change, ecological interactions, dispersal, and other factors. The differences in chlorophyll temporal variability observed at stationary locations (Eulerian perspective) or following water parcels (Lagrangian perspective) are poorly understood. Here we contrasted the temporal variability of ocean chlorophyll in these two observational perspectives, using global drifter trajectories and satellite chlorophyll to generate matched pairs of Eulerian‐Lagrangian time series. We found that for most ocean locations, chlorophyll variances measured in Eulerian and Lagrangian perspectives are not statistically different. In high latitude areas, the two perspectives may capture similar variability due to the large spatial scale of chlorophyll patches. In localized regions of the ocean, however, chlorophyll variability measured in these two perspectives may significantly differ. For example, in some western boundary currents, temporal chlorophyll variability in the Lagrangian perspective was greater than in the Eulerian perspective. In these cases, the observing platform travels rapidly across strong environmental gradients and constrained by the shelf topography, potentially leading to greater Lagrangian variability in chlorophyll. In contrast, we found that Eulerian chlorophyll variability exceeded Lagrangian variability in some key upwelling zones and boundary current extensions. In these cases, variability in the nutrient supply may generate intermittent chlorophyll anomalies in the Eulerian perspective, while the Lagrangian perspective sees the transport of such anomalies off‐shore. These findings aid with the interpretation of chlorophyll time series from different sampling methodologies, inform observational network design, and guide validation of marine ecosystem models. 

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3. Guiding-centre Lagrangian and quasi-symmetry

   https://doi.org/10.1017/s0022377825000133  

   Jacobson, Ted (April 2025, Journal of Plasma Physics)

   A charged particle in a suitably strong magnetic field spirals along the field lines while slowly drifting transversely. This note provides a brief derivation of an effective Lagrangian formulation for the guiding-centre approximation that captures this dynamics without resolving the gyro motion. It also explains how the effective Lagrangian may, for special magnetic fields, admit a ‘quasi-symmetry’ which can give rise to a conserved quantity helpful for plasma confinement in fields lacking a geometric isometry. The aim of this note is to offer a pedagogical introduction and some perspectives on this well-established subject. 

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4. A finiteness theorem for Lagrangian fibrations

   https://doi.org/10.1090/jag/673  

   Sawon, Justin (July 2016, Journal of Algebraic Geometry)

   We consider (holomorphic) Lagrangian fibrations $\mathrm{\pi <\#comment/>}:X\to <\#comment/>{\mathbb{P}}^n$that satisfy some natural hypotheses. We prove that there are only finitely many such Lagrangian fibrations up to deformation. 

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5. Lagrangian skeleta and plane curve singularities

   https://doi.org/10.1007/s11784-022-00939-8  

   Casals, Roger (June 2022, Journal of Fixed Point Theory and Applications)

   Abstract We construct closed arboreal Lagrangian skeleta associated to links of isolated plane curve singularities. This yields closed Lagrangian skeleta for Weinstein pairs $$(\mathbb {C}^2,\Lambda )$$ ( C 2 , Λ ) and Weinstein 4-manifolds $$W(\Lambda )$$ W ( Λ ) associated to max-tb Legendrian representatives of algebraic links $$\Lambda \subseteq (\mathbb {S}^3,\xi _\text {st})$$ Λ ⊆ ( S 3 , ξ st ) . We provide computations of Legendrian and Weinstein invariants, and discuss the contact topological nature of the Fomin–Pylyavskyy–Shustin–Thurston cluster algebra associated to a singularity. Finally, we present a conjectural ADE-classification for Lagrangian fillings of certain Legendrian links and list some related problems. 

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