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1. 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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2. Type systems and maximal subgroups of Thompson’s group 𝑉

   https://doi.org/10.1090/btran/204  

   Belk, James; Bleak, Collin; Quick, Martyn; Skipper, Rachel (April 2025, Transactions of the American Mathematical Society, Series B)

   We introduce the concept of a type system  $\mathcal{P}$, that is, a partition on the set of finite words over the alphabet  $\{0,1\}$compatible with the partial action of Thompson’s group  $V$, and associate a subgroup  ${\mathrm{Stab}}_V<\#comment/>\left(\mathcal{P}\right)$of  $V$. We classify the finite simple type systems and show that the stabilizers of various simple type systems, including all finite simple type systems, are maximal subgroups of  $V$. We also find an uncountable family of pairwise nonisomorphic maximal subgroups of  $V$. These maximal subgroups occur as stabilizers of infinite simple type systems and have not been described in previous literature: specifically, they do not arise as stabilizers in $V$of finite sets of points in Cantor space. Finally, we show that two natural conditions on subgroups of $V$(both related to primitivity) are each satisfied only by $V$itself, giving new ways to recognise when a subgroup of $V$is not actually proper. 

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3. The structure of arbitrary Conze–Lesigne systems

   https://doi.org/10.1090/cams/31  

   Jamneshan, Asgar; Shalom, Or; Tao, Terence (February 2024, Communications of the American Mathematical Society)

   Let $\mathrm{\Gamma <\#comment/>}$be a countable abelian group. An (abstract) $\mathrm{\Gamma <\#comment/>}$-system $\mathrm{X}$- that is, an (abstract) probability space equipped with an (abstract) probability-preserving action of $\mathrm{\Gamma <\#comment/>}$- is said to be aConze–Lesigne systemif it is equal to its second Host–Kra–Ziegler factor ${\mathrm{Z}}^2\left(\mathrm{X}\right)$. The main result of this paper is a structural description of such Conze–Lesigne systems for arbitrary countable abelian $\mathrm{\Gamma <\#comment/>}$, namely that they are the inverse limit of translational systems $G_n/{\mathrm{\Lambda <\#comment/>}}_n$arising from locally compact nilpotent groups $G_n$of nilpotency class $2$, quotiented by a lattice ${\mathrm{\Lambda <\#comment/>}}_n$. Results of this type were previously known when $\mathrm{\Gamma <\#comment/>}$was finitely generated, or the product of cyclic groups of prime order. In a companion paper, two of us will apply this structure theorem to obtain an inverse theorem for the Gowers $U^3\left(G\right)$norm for arbitrary finite abelian groups $G$. 

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4. Residual intersections and linear powers

   https://doi.org/10.1090/btran/127  

   Eisenbud, David; Huneke, Craig; Ulrich, Bernd (October 2023, Transactions of the American Mathematical Society, Series B)

   If $I$is an ideal in a Gorenstein ring $S$, and $S/I$is Cohen-Macaulay, then the same is true for any linked ideal $I^{\prime }$; but such statements hold for residual intersections of higher codimension only under restrictive hypotheses, not satisfied even by ideals as simple as the ideal $L_n$of minors of a generic $2\times <\#comment/>n$matrix when $n>3$. In this paper we initiate the study of a different sort of Cohen-Macaulay property that holds for certain general residual intersections of the maximal (interesting) codimension, one less than the analytic spread of $I$. For example, suppose that $K$is the residual intersection of $L_n$by $2n-<\#comment/>4$general quadratic forms in $L_n$. In this situation we analyze $S/K$and show that $I^{n-<\#comment/>3}\left(S/K\right)$is a self-dual maximal Cohen-Macaulay $S/K$-module with linear free resolution over $S$. The technical heart of the paper is a result about ideals of analytic spread 1 whose high powers are linearly presented. 

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5. Products of normal subsets

   https://doi.org/10.1090/tran/8960  

   Larsen, Michael; Shalev, Aner; Tiep, Pham (February 2024, Transactions of the American Mathematical Society)

   In this paper we consider which families of finite simple groups $G$have the property that for each $\mathrm{ϵ<\#comment/>}>0$there exists $N>0$such that, if $|G|\ge <\#comment/>N$and $S,T$are normal subsets of $G$with at least $\mathrm{ϵ<\#comment/>}|G|$elements each, then every non-trivial element of $G$is the product of an element of $S$and an element of $T$. We show that this holds in a strong and effective sense for finite simple groups of Lie type of bounded rank, while it does not hold for alternating groups or groups of the form ${\mathrm{P}\mathrm{S}\mathrm{L}}_n\left(q\right)$where $q$is fixed and $n\to <\#comment/>\mathrm{\infty <\#comment/>}$. However, in the case $S=T$and $G$alternating this holds with an explicit bound on $N$in terms of $\mathrm{ϵ<\#comment/>}$. Related problems and applications are also discussed. In particular we show that, if $w_1,w_2$are non-trivial words, $G$is a finite simple group of Lie type of bounded rank, and for $g\in <\#comment/>G$, $P_{w_1\left(G\right),w_2\left(G\right)}\left(g\right)$denotes the probability that $g_1{g}_2=g$where $g_i\in <\#comment/>w_i\left(G\right)$are chosen uniformly and independently, then, as $|G|\to <\#comment/>\mathrm{\infty <\#comment/>}$, the distribution $P_{w_1\left(G\right),w_2\left(G\right)}$tends to the uniform distribution on $G$with respect to the $L^{\mathrm{\infty <\#comment/>}}$norm. 

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