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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 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. On the Concentration of the Chromatic Number of Random Graphs

   https://doi.org/10.37236/11638  

   Surya, Erlang; Warnke, Lutz (January 2024, The Electronic Journal of Combinatorics)

   Shamir and Spencer proved in the 1980s that the chromatic number of the binomial random graph $$G_{n,p}$$ is concentrated in an interval of length at most $$\omega\sqrt{n}$$, and in the 1990s Alon showed that an interval of length $$\omega\sqrt{n}/\log n$$ suffices for constant edge-probabilities $$p\in (0,1)$$. We prove a similar logarithmic improvement of the Shamir-Spencer concentration results for the sparse case $${p=p(n) \to 0}$$, and uncover a surprising concentration `jump' of the chromatic number in the very dense case $${p=p(n) \to 1}$$. 

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4. The rational homotopy type of the projectivization of the tangent bundle of complex projective spaces

   https://doi.org/10.1007/s13370-025-01393-8  

   Ndlovu, Meshach; Gatsinzi, Jean Baptiste (December 2025, Afrika Matematika)

   We determine the rational homotopy type of the total space of the projectivization of the complex tangent bundle τ : Cⁿ −→ E −→ CPⁿ. We show that the total space P(E) of the projectivization bundle P (τ ) : CPⁿ−¹ −→ P (E) −→ CPⁿ has the rational homotopy type of U (n + 1)/U (1) × U (1) × U (n − 1), where U(k) is the unitary group. Not Available 

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5. The determination of leaf size on the basis of developmental traits

   https://doi.org/10.1111/nph.20461  

   Ma, Zeqing; Buckley, Thomas N; Sack, Lawren (April 2025, New Phytologist)

   Summary Mature leaf area (LA) is a showcase of diversity – varying enormously within and across species, and associated with the productivity and distribution of plants and ecosystems. Yet, it remains unclear how developmental processes determine variation in LA.We introduce a mathematical framework pinpointing the origin of variation in LA by quantifying six epidermal ‘developmental traits’: initial mean cell size and number (approximating values within the leaf primordium), and the maximum relative rates and durations of cell proliferation and expansion until leaf maturity. We analyzed a novel database of developmental trajectories of LA and epidermal anatomy, representing 12 eudicotyledonous species and 52 Arabidopsis experiments.Within and across species, mean primordium cell number and maximum relative cell proliferation rate were the strongest developmental determinants of LA. Trade‐offs between developmental traits, consistent with evolutionary and metabolic scaling theory, strongly constrain LA variation. These include trade‐offs between primordium cell number vs cell proliferation, primordium mean cell size vs cell expansion, and the durations vs maximum relative rates of cell proliferation and expansion. Mutant and wild‐type comparisons showed these trade‐offs have a genetic basis in Arabidopsis.Analyses of developmental traits underlying LA and its diversification highlight mechanisms for leaf evolution, and opportunities for breeding trait shifts. 

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