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1. 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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2. Torsion for CM elliptic curves defined over number fields of degree 2𝑝

   https://doi.org/10.1090/proc/16170  

   Bourdon, Abbey; Chaos, Holly Paige (March 2023, Proceedings of the American Mathematical Society)

   For a prime number $p$, we characterize the groups that may arise as torsion subgroups of an elliptic curve with complex multiplication defined over a number field of degree $2p$. In particular, our work shows that a classification in the strongest sense is tied to determining whether there exist infinitely many Sophie Germain primes. 

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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. Quantitative structure of stable sets in arbitrary finite groups

   https://doi.org/10.1090/proc/15479  

   Conant, Gabriel (September 2021, Proceedings of the American Mathematical Society)

   We show that a $k$-stable set in a finite group can be approximated, up to given error $\mathrm{ϵ<\#comment/>}>0$, by left cosets of a subgroup of index ${\mathrm{ϵ<\#comment/>}}^{\text{-}{O}_k\left(1\right)}$. This improves the bound in a similar result of Terry and Wolf on stable arithmetic regularity in finite abelian groups, and leads to a quantitative account of work of the author, Pillay, and Terry on stable sets in arbitrary finite groups. We also prove an analogous result for finite stable sets of small tripling in arbitrary groups, which provides a quantitative version of recent work by Martin-Pizarro, Palacín, and Wolf. Our proofs use results on VC-dimension, and a finitization of model-theoretic techniques from stable group theory. 

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5. The unbounded denominators conjecture

   https://doi.org/10.1090/jams/1053  

   Calegari, Frank; Dimitrov, Vesselin; Tang, Yunqing (February 2025, Journal of the American Mathematical Society)

   We prove the unbounded denominators conjecture in the theory of noncongruence modular forms for finite index subgroups of ${\mathrm{SL}}_2<\#comment/>\left(\mathbf{Z}\right)$. Our result includes also Mason’s generalization of the original conjecture to the setting of vector-valued modular forms, thereby supplying a new path to the congruence property in rational conformal field theory. The proof involves a new arithmetic holonomicity bound of a potential-theoretic flavor, together with Nevanlinna second main theorem, the congruence subgroup property of ${\mathrm{SL}}_2<\#comment/>\left(\mathbf{Z}\right[1/p\left]\right)$, and a close description of the Fuchsian uniformization $D(0,1)/{\mathrm{\Gamma <\#comment/>}}_N$of the Riemann surface $\mathbf{C}\setminus <\#comment/>{\mathrm{\mu <\#comment/>}}_N$. 

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