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1. On the structure of Besse convex contact spheres

   https://doi.org/10.1090/tran/8836  

   Mazzucchelli, Marco; Radeschi, Marco (March 2023, Transactions of the American Mathematical Society)

   We consider convex contact spheres Y Y all of whose Reeb orbits are closed. Any such Y Y admits a stratification by the periods of closed Reeb orbits. We show that Y Y “resembles” a contact ellipsoid: any stratum of Y Y is an integral homology sphere, and the sequence of Ekeland-Hofer spectral invariants of Y Y coincides with the full sequence of action values, each one repeated according to its multiplicity. 

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2. Two-Sided Kirszbraun Theorem

   https://doi.org/10.4230/LIPIcs.SoCG.2021.13  

   Backurs, Arturs; Mahabadi, Sepideh; Makarychev, Konstantin; Makarychev, Yury (June 2021, Leibniz international proceedings in informatics)

   Buchin, Kevin; Colin de Verdiere, Eric (Ed.)

   In this paper, we prove a two-sided variant of the Kirszbraun theorem. Consider an arbitrary subset X of Euclidean space and its superset Y. Let f be a 1-Lipschitz map from X to ℝ^m. The Kirszbraun theorem states that the map f can be extended to a 1-Lipschitz map ̃ f from Y to ℝ^m. While the extension ̃ f does not increase distances between points, there is no guarantee that it does not decrease distances significantly. In fact, ̃ f may even map distinct points to the same point (that is, it can infinitely decrease some distances). However, we prove that there exists a (1 + ε)-Lipschitz outer extension f̃:Y → ℝ^{m'} that does not decrease distances more than "necessary". Namely, ‖f̃(x) - f̃(y)‖ ≥ c √{ε} min(‖x-y‖, inf_{a,b ∈ X} (‖x - a‖ + ‖f(a) - f(b)‖ + ‖b-y‖)) for some absolutely constant c > 0. This bound is asymptotically optimal, since no L-Lipschitz extension g can have ‖g(x) - g(y)‖ > L min(‖x-y‖, inf_{a,b ∈ X} (‖x - a‖ + ‖f(a) - f(b)‖ + ‖b-y‖)) even for a single pair of points x and y. In some applications, one is interested in the distances ‖f̃(x) - f̃(y)‖ between images of points x,y ∈ Y rather than in the map f̃ itself. The standard Kirszbraun theorem does not provide any method of computing these distances without computing the entire map ̃ f first. In contrast, our theorem provides a simple approximate formula for distances ‖f̃(x) - f̃(y)‖. 

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3. Unique structure and positive selection promote the rapid divergence of Drosophila Y chromosomes

   https://doi.org/10.7554/eLife.75795  

   Chang, Ching-Ho; Gregory, Lauren E; Gordon, Kathleen E; Meiklejohn, Colin D; Larracuente, Amanda M (January 2022, eLife)

   Y chromosomes across diverse species convergently evolve a gene-poor, heterochromatic organization enriched for duplicated genes, LTR retrotransposons, and satellite DNA. Sexual antagonism and a loss of recombination play major roles in the degeneration of young Y chromosomes. However, the processes shaping the evolution of mature, already degenerated Y chromosomes are less well-understood. Because Y chromosomes evolve rapidly, comparisons between closely related species are particularly useful. We generated de novo long-read assemblies complemented with cytological validation to reveal Y chromosome organization in three closely related species of the Drosophila simulans complex, which diverged only 250,000 years ago and share >98% sequence identity. We find these Y chromosomes are divergent in their organization and repetitive DNA composition and discover new Y-linked gene families whose evolution is driven by both positive selection and gene conversion. These Y chromosomes are also enriched for large deletions, suggesting that the repair of double-strand breaks on Y chromosomes may be biased toward microhomology-mediated end joining over canonical non-homologous end-joining. We propose that this repair mechanism contributes to the convergent evolution of Y chromosome organization across organisms. 

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4. Gene-Level, but Not Chromosome-Wide, Divergence between a Very Young House Fly Proto-Y Chromosome and Its Homologous Proto-X Chromosome

   https://doi.org/10.1093/molbev/msaa250  

   Son, Jae Hak; Meisel, Richard P (September 2020, Molecular Biology and Evolution)

   null (Ed.)

   Abstract X and Y chromosomes are usually derived from a pair of homologous autosomes, which then diverge from each other over time. Although Y-specific features have been characterized in sex chromosomes of various ages, the earliest stages of Y chromosome evolution remain elusive. In particular, we do not know whether early stages of Y chromosome evolution consist of changes to individual genes or happen via chromosome-scale divergence from the X. To address this question, we quantified divergence between young proto-X and proto-Y chromosomes in the house fly, Musca domestica. We compared proto-sex chromosome sequence and gene expression between genotypic (XY) and sex-reversed (XX) males. We find evidence for sequence divergence between genes on the proto-X and proto-Y, including five genes with mitochondrial functions. There is also an excess of genes with divergent expression between the proto-X and proto-Y, but the number of genes is small. This suggests that individual proto-Y genes, but not the entire proto-Y chromosome, have diverged from the proto-X. We identified one gene, encoding an axonemal dynein assembly factor (which functions in sperm motility), that has higher expression in XY males than XX males because of a disproportionate contribution of the proto-Y allele to gene expression. The upregulation of the proto-Y allele may be favored in males because of this gene’s function in spermatogenesis. The evolutionary divergence between proto-X and proto-Y copies of this gene, as well as the mitochondrial genes, is consistent with selection in males affecting the evolution of individual genes during early Y chromosome evolution. 

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5. Fillable contact structures from positive surgery

   https://doi.org/10.1090/btran/200  

   Mark, Thomas; Tosun, Bülent (August 2024, Transactions of the American Mathematical Society, Series B)

   We consider the question of when the operation of contact surgery with positive surgery coefficient, along a knot $K$in a contact 3-manifold $Y$, gives rise to a weakly fillable contact structure. We show that this happens if and only if $Y$itself is weakly fillable, and $K$is isotopic to the boundary of a properly embedded symplectic disk inside a filling of $Y$. Moreover, if $Y^{\prime }$is a contact manifold arising from positive contact surgery along $K$, then any filling of $Y^{\prime }$is symplectomorphic to the complement of a suitable neighborhood of such a disk in a filling of $Y$. Using this result we deduce several necessary conditions for a knot in the standard 3-sphere to admit a fillable positive surgery, such as quasipositivity and equality between the slice genus and the 4-dimensional clasp number, and we give a characterization of such knots in terms of a quasipositive braid expression. We show that knots arising as the closure of a positive braid always admit a fillable positive surgery, as do knots that have lens space surgeries, and suitable satellites of such knots. In fact the majority of quasipositive knots with up to 10 crossings admit a fillable positive surgery. On the other hand, in general, (strong) quasipositivity, positivity, or Lagrangian fillability need not imply a knot admits a fillable positive contact surgery. 

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