More Like this

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. 

   more » « less
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  ${\mathbb{F}}_2$. 

   more » « less
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. 

   more » « less
4. 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. 

   more » « less
5. An optimal approximation problem for free polynomials

   https://doi.org/10.1090/proc/16474  

   Arora, Palak; Augat, Meric; Jury, Michael; Sargent, Meredith (November 2023, Proceedings of the American Mathematical Society)

   Motivated by recent work on optimal approximation by polynomials in the unit disk, we consider the following noncommutative approximation problem: for a polynomial $f$in $d$freely noncommuting arguments, find a free polynomial $p_n$, of degree at most $n$, to minimize $c_n≔\Vert <\#comment/>p_{n}f-<\#comment/>1{\Vert <\#comment/>}^2$. (Here the norm is the ${\mathrm{\ell <\#comment/>}}^2$norm on coefficients.) We show that $c_n\to <\#comment/>0$if and only if $f$is nonsingular in a certain nc domain (the row ball), and prove quantitative bounds. As an application, we obtain a new proof of the characterization of polynomials cyclic for the $d$-shift. 

   more » « less