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   https://doi.org/10.1007/s00493-024-00128-w  

   Li, Lina; McKinley, Gweneth; Park, Jinyoung (February 2025, Combinatorica)
2. THE BOUNDARY OF THE p -RANK STRATUM OF THE MODULI SPACE OF CYCLIC COVERS OF THE PROJECTIVE LINE

   https://doi.org/10.1017/nmj.2022.12  

   OZMAN, EKIN; PRIES, RACHEL; WEIR, COLIN (December 2022, Nagoya Mathematical Journal)

   Abstract. We study the p-rank stratification of the moduli space of cyclic degree ! covers of the projective line in characteristic p for distinct primes p and !. The main result is about the intersection of the p-rank 0 stratum with the boundary of the moduli space of curves. When ! = 3 and p ≡ 2 mod 3 is an odd prime, we prove that there exists a smooth trielliptic curve in characteristic p, for every genus g, signature type (r,s), and p-rank f satisfying the clear necessary conditions. 

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3. The Grayness of the Origin of Life

   https://doi.org/10.3390/life11060498  

   Smith, Hillary H.; Hyde, Andrew S.; Simkus, Danielle N.; Libby, Eric; Maurer, Sarah E.; Graham, Heather V.; Kempes, Christopher P.; Sherwood Lollar, Barbara; Chou, Luoth; Ellington, Andrew D.; et al (June 2021, Life)

   null (Ed.)

   In the search for life beyond Earth, distinguishing the living from the non-living is paramount. However, this distinction is often elusive, as the origin of life is likely a stepwise evolutionary process, not a singular event. Regardless of the favored origin of life model, an inherent “grayness” blurs the theorized threshold defining life. Here, we explore the ambiguities between the biotic and the abiotic at the origin of life. The role of grayness extends into later transitions as well. By recognizing the limitations posed by grayness, life detection researchers will be better able to develop methods sensitive to prebiotic chemical systems and life with alternative biochemistries. 

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4. 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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5. Capacity of the range of random walk: The law of the iterated logarithm

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   Dembo, Amir; Okada, Izumi (September 2024, The Annals of Probability)