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1. Propagation of singularities by Osgood vector fields and for 2D inviscid incompressible fluids

   https://doi.org/10.1007/s00208-022-02498-2  

   Drivas, Theodore D.; Elgindi, Tarek M.; La, Joonhyun (October 2022, Mathematische Annalen)

   Abstract We show that certain singular structures (Hölderian cusps and mild divergences) are transported by the flow of homeomorphisms generated by an Osgood velocity field. The structure of these singularities is related to the modulus of continuity of the velocity and the results are shown to be sharp in the sense that slightly more singular structures cannot generally be propagated. For the 2D Euler equation, we prove that certain singular structures are preserved by the motion, e.g. a system of$$\log \log _+(1/|x|)$$ $log{log}_{+}(1/|x\left|\right)$vortices (and those that are slightly less singular) travel with the fluid in a nonlinear fashion, up to bounded perturbations. We also give stability results for weak Euler solutions away from their singular set. 

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2. Adams’ cobar construction as a monoidal -coalgebra model of the based loop space

   https://doi.org/10.1017/fms.2024.50  

   Medina-Mardones, Anibal M; Rivera, Manuel (January 2024, Forum of Mathematics, Sigma)

   Abstract We prove that the classical map comparing Adams’ cobar construction on the singular chains of a pointed space and the singular cubical chains on its based loop space is a quasi-isomorphism preserving explicitly defined monoidal$$E_\infty $$-coalgebra structures. This contribution extends to its ultimate conclusion a result of Baues, stating that Adams’ map preserves monoidal coalgebra structures. 

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3. Singular Abreu equations and linearized Monge–Ampère equations with drifts

   https://doi.org/10.4171/JEMS/1548  

   Kim, Young Ho; Le, Nam Q; Wang, Ling; Zhou, Bin (October 2024, Journal of the European Mathematical Society)

   We study the solvability of singular Abreu equations which arise in the approximation of convex functionals subject to a convexity constraint. Previous works established the solvability of their second boundary value problems either in two dimensions, or in higher dimensions under either a smallness condition or a radial symmetry condition. Here, we solve the higher-dimensional case by transforming singular Abreu equations into linearized Monge–Ampère equations with drifts. We establish global Hölder estimates for linearized Monge–Ampère equations with drifts under suitable hypotheses, and then apply them to prove the regularity and solvability of the second boundary value problem for singular Abreu equations in higher dimensions. Many cases with general right-hand side are also discussed. 

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4. Topological solitons, cholesteric fingers and singular defect lines in Janus liquid crystal shells

   https://doi.org/10.1039/C9SM02033K  

   Durey, Guillaume; Sohn, Hayley R.; Ackerman, Paul J.; Brasselet, Etienne; Smalyukh, Ivan I.; Lopez-Leon, Teresa (March 2020, Soft Matter)

   Topological solitons are non-singular but topologically nontrivial structures in fields, which have fundamental significance across various areas of physics, similar to singular defects. Production and observation of singular and solitonic topological structures remain a complex undertaking in most branches of science – but in soft matter physics, they can be realized within the director field of a liquid crystal. Additionally, it has been shown that confining liquid crystals to spherical shells using microfluidics resulted in a versatile experimental platform for the dynamical study of topological transformations between director configurations. In this work, we demonstrate the triggered formation of topological solitons, cholesteric fingers, singular defect lines and related structures in liquid crystal shells. We show that to accommodate these objects, shells must possess a Janus nature, featuring both twisted and untwisted domains. We report the formation of linear and axisymmetric objects, which we identify as cholesteric fingers and skyrmions or elementary torons, respectively. We then take advantage of the sensitivity of shells to numerous external stimuli to induce dynamical transitions between various types of structures, allowing for a richer phenomenology than traditional liquid crystal cells with solid flat walls. Using gradually more refined experimental techniques, we induce the targeted transformation of cholesteric twist walls and fingers into skyrmions and elementary torons. We capture the different stages of these director transformations using numerical simulations. Finally, we uncover an experimental mechanism to nucleate arrays of axisymmetric structures on shells, thereby creating a system of potential interest for tackling crystallography studies on curved spaces. 

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5. Tight contact structures on some families of small Seifert fiber spaces

   https://doi.org/10.1007/s10474-024-01444-9  

   Wan, S (June 2024, Acta Mathematica Hungarica)

   Abstract SupposeKis a knot in a 3-manifoldY, and thatYadmits a pair of distinct contact structures. Assume thatKhas Legendrian representatives in each of these contact structures, such that the corresponding Thurston-Bennequin framings are equivalent. This paper provides a method to prove that the contact structures resulting from Legendrian surgery along these two representatives remain distinct. Applying this method to the situation where the starting manifold is$$-\Sigma(2,3,6m+1)$$ $-\Sigma (2,3,6m+1)$and the knot is a singular fiber, together with convex surface theory we can classify the tight contact structures on certain families of Seifert fiber spaces. 

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