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1. The eigensplitting of the fiber of the cyclotomic trace for the sphere spectrum

   https://doi.org/10.1090/tran/8822  

   Blumberg, Andrew; Mandell, Michael (January 2022, Transactions of the American Mathematical Society)

   Let p ∈ Z p\in {\mathbb {Z}} be an odd prime. We show that the fiber sequence for the cyclotomic trace of the sphere spectrum S {\mathbb {S}} admits an “eigensplitting” that generalizes known splittings on K K -theory and T C TC . We identify the summands in the fiber as the covers of Z p {\mathbb {Z}}_{p} -Anderson duals of summands in the K ( 1 ) K(1) -localized algebraic K K -theory of Z {\mathbb {Z}} . Analogous results hold for the ring Z {\mathbb {Z}} where we prove that the K ( 1 ) K(1) -localized fiber sequence is self-dual for Z p {\mathbb {Z}}_{p} -Anderson duality, with the duality permuting the summands by i ↦ p − i i\mapsto p-i (indexed mod p − 1 p-1 ). We explain an intrinsic characterization of the summand we call Z Z in the splitting T C ( Z ) p ∧ ≃ j ∨ Σ j ′ ∨ Z TC({\mathbb {Z}})^{\wedge }_{p}\simeq j \vee \Sigma j’\vee Z in terms of units in the p p -cyclotomic tower of Q p {\mathbb {Q}}_{p} . 

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2. The Origin of Universality in the Inner Edges of Planetary Systems

   https://doi.org/10.3847/2041-8213/acdb5d  

   Batygin, Konstantin; Adams, Fred C; Becker, Juliette (July 2023, The Astrophysical Journal Letters)

   Abstract The characteristic orbital period of the innermost objects within the galactic census of planetary and satellite systems appears to be nearly universal, withPon the order of a few days. This paper presents a theoretical framework that provides a simple explanation for this phenomenon. By considering the interplay between disk accretion, magnetic field generation by convective dynamos, and Kelvin–Helmholtz contraction, we derive an expression for the magnetospheric truncation radius in astrophysical disks and find that the corresponding orbital frequency is independent of the mass of the host body. Our analysis demonstrates that this characteristic frequency corresponds to a period ofP∼ 3 days although intrinsic variations in system parameters are expected to introduce a factor of a ∼2–3 spread in this result. Standard theory of orbital migration further suggests that planets should stabilize at an orbital period that exceeds disk truncation by a small margin. Cumulatively, our findings predict that the periods of close-in bodies should spanP∼ 2–12 days—a range that is consistent with observations. 

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3. The dynamics and changes of the world’s monsoons

   https://doi.org/10.1063/PT.3.5308  

   Biasutti, Michela; Ting, Mingfang; Hill, Spencer A. (September 2023, Physics Today)

   The seasonal rainy phase observed in many places across Earth is shaping the climate and is being changed by global climate trends. 

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4. A short proof of the existence of the injective envelope of an operator space

   https://doi.org/10.1090/proc/17412  

   Sinclair, Thomas (October 2025, Proceedings of the American Mathematical Society)

   We use Ellis’ lemma to give a simple proof of the existence of the injective envelope of an operator space first shown by work of Hamana and Ruan. 

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5. The Role of Gene Flow in the Diversification of the Monkey Treefrog Complex Across the South American Dry Diagonal

   https://doi.org/10.1111/zsc.70046  

   Camurugi, Felipe; São_Pedro, Vinícius A; Magalhães, Felipe; Gehara, Marcelo; Oliveira, Eliana F; Costa, Gabriel C; Burbrink, Frank T; Myers, Edward A; Colli, Guarino R; Rodrigues, Miguel T; et al (July 2026, Zoologica Scripta)

   ABSTRACT Understanding Neotropical megadiversity remains challenging due to fundamental taxonomic issues, including identifying and describing cryptic species and their distribution, and the limited knowledge of key factors driving biological diversification. Such challenges are especially prominent in diverse clades with high levels of cryptic species, such as many Neotropical frogs. Herein, we conduct a mitochondrial phylogenetic study on monkey treefrogs in the genusPithecopusand a multilocus analysis of theP. hypochondrialisspecies group throughout its distribution in the South American dry diagonal (DD). Our goals were to infer the main lineages and identify historical factors that led to their divergence. Among nine currently recognised species, we found 14 well‐structured mitochondrial lineages in two main clades, which diversified during the Paleogene/Neogene transition. The lowland clade is widely distributed in the DD, and our multilocus analysis suggests it comprises seven geographically structured lineages:P. azureus, two lineages ofP. hypochondrialis, two species and one lineage for theP. nordestinuscomplex, andPithecopussp. Considering the geomorphological history of the region, diversification may have been promoted by allopatric speciation during Miocene paleogeographic events, especially the Brazilian Plateau compartmentalization and fluctuations in the São Francisco River water volume. Furthermore, gene flow between some lineages may be explained by niche conservatism and demographic expansions during the late Pleistocene, a period marked by climatic fluctuations and biome shifts. Our results reinforce the importance of both Neogene tectonic activity and Quaternary climatic fluctuations in shaping the Neotropical biota. 

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