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1. Naturality and innerness for morphisms of compact groups and (restricted) Lie algebras

   https://doi.org/10.1090/bproc/164  

   Chirvasitu, Alexandru (July 2024, Proceedings of the American Mathematical Society, Series B)

   An extended derivation (endomorphism) of a (restricted) Lie algebra $L$ is an assignment of a derivation (respectively) of $L^{\prime }$ for any (restricted) Lie morphism $f:L\to <\#comment/>L^{\prime }$ , functorial in $f$ in the obvious sense. We show that (a) the only extended endomorphisms of a restricted Lie algebra are the two obvious ones, assigning either the identity or the zero map of $L^{\prime }$ to every $f$ ; and (b) if $L$ is a Lie algebra in characteristic zero or a restricted Lie algebra in positive characteristic, then $L$ is in canonical bijection with its space of extended derivations (so the latter are all, in a sense, inner). These results answer a number of questions of G. Bergman. In a similar vein, we show that the individual components of an extended endomorphism of a compact connected group are either all trivial or all inner automorphisms. 

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2. Every finite abelian group is the group of rational points of an ordinary abelian variety over 𝔽₂, 𝔽₃ and 𝔽₅

   https://doi.org/10.1090/proc/16127  

   Marseglia, Stefano; Springer, Caleb (February 2023, Proceedings of the American Mathematical Society)

   We show that every finite abelian group occurs as the group of rational points of an ordinary abelian variety over ${\mathbb{F}}_2$ , ${\mathbb{F}}_3$ and ${\mathbb{F}}_5$ . We produce partial results for abelian varieties over a general finite field  ${\mathbb{F}}_q$ . In particular, we show that certain abelian groups cannot occur as groups of rational points of abelian varieties over ${\mathbb{F}}_q$ when $q$ is large. Finally, we show that every finite cyclic group arises as the group of rational points of infinitely many simple abelian varieties over  ${\mathbb{F}}_2$ . 

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3. Positroid Catalan numbers

   https://doi.org/10.1090/cams/33  

   Galashin, Pavel; Lam, Thomas (April 2024, Communications of the American Mathematical Society)

   Given a (bounded affine) permutation $f$ , we study thepositroid Catalan number $C_f$ defined to be the torus-equivariant Euler characteristic of the associated open positroid variety. We introduce a class ofrepetition-free permutationsand show that the corresponding positroid Catalan numbers count Dyck paths avoiding a convex subset of the rectangle. We show that any convex subset appears in this way. Conjecturally, the associated $q,t$ -polynomials coincide with thegeneralized $q,t$ -Catalan numbersthat recently appeared in relation to the shuffle conjecture, flag Hilbert schemes, and Khovanov–Rozansky homology of Coxeter links. 

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4. Stable sheaf cohomology on flag varieties

   https://doi.org/10.1090/jams/1071  

   Raicu, Claudiu; VandeBogert, Keller (March 2026, Journal of the American Mathematical Society)

   We prove an effective stabilization result for the sheaf cohomology groups of line bundles on flag varieties $F{l}_n$ parametrizing complete flags in ${\mathbf{k}}^n$ , as well as for the sheaf cohomology groups of polynomial functors applied to the cotangent sheaf $\mathrm{\Omega <\#comment/>}$ on projective space. In characteristic zero, these are natural consequences of the Borel–Weil–Bott theorem, but in characteristic $p>0$ they are non-trivial. Unlike many important contexts in modular representation theory, where the prime characteristic $p$ is assumed to be large relative to $n$ , in our study we fix $p$ and let $n⟶<\#comment/>\mathrm{\infty <\#comment/>}$ . We illustrate the general theory by providing explicit stable cohomology calculations in a number of cases of interest. Our examples yield cohomology groups where the number of indecomposable summands has super-polynomial growth, and also show that the cohomological degrees where non-vanishing occurs do not form a connected interval. In the case of polynomial functors of $\mathrm{\Omega <\#comment/>}$ , we prove a Künneth formula for stable cohomology, and show the invariance of stable cohomology under Frobenius, which combined with the Steinberg tensor product theorem yields calculations of stable cohomology for an interesting class of simple polynomial functors arising in the work of Doty. The results in the special case of symmetric powers of $\mathrm{\Omega <\#comment/>}$ provide a nice application to commutative algebra, yielding a sharp vanishing result for Koszul modules of finite length in all characteristics. 

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5. A lower bound on end-periodic stretch factors

   https://doi.org/10.1090/proc/17304  

   Loving, Marissa; Wu, Chenxi (July 2025, Proceedings of the American Mathematical Society)

   Given an end-periodic homeomorphism $f:S\to <\#comment/>S$ we give a lower bound on the Handel–Miller stretch factor of $f$ in terms of thecore characteristic of $f$ , which is a measure of topological complexity for an end-periodic homeomorphism. We also show that the growth rate of this bound is sharp. 

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