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1. 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$\epsilon >0$, by left cosets of a subgroup of index$\epsilon ^{\text {-} O_k(1)}$. 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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2. 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$\Gamma$be a countable abelian group. An (abstract)$\Gamma$-system$\mathrm {X}$- that is, an (abstract) probability space equipped with an (abstract) probability-preserving action of$\Gamma$- is said to be aConze–Lesigne systemif it is equal to its second Host–Kra–Ziegler factor$\mathrm {Z}^2(\mathrm {X})$. The main result of this paper is a structural description of such Conze–Lesigne systems for arbitrary countable abelian$\Gamma$, namely that they are the inverse limit of translational systems$G_n/\Lambda _n$arising from locally compact nilpotent groups$G_n$of nilpotency class$2$, quotiented by a lattice$\Lambda _n$. Results of this type were previously known when$\Gamma$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$U^3(G)$norm for arbitrary finite abelian groups$G$.

    

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3. Characterization of positive definite, radial functions on free groups

   https://doi.org/10.1090/bproc/158  

   Chuah, Chian Yeong ; Liu, Zhenchuan ; Mei, Tao ( April 2023 , Proceedings of the American Mathematical Society, Series B)

   This article studies the properties of positive definite, radial functions on free groups following the work of Haagerup and Knudby [Proc. Amer. Math. Soc. 143 (2015), pp. 1477–1489]. We obtain characterizations of radial functions with respect to the$\ell ^{2}$length on the free groups with infinite generators and the characterization of the positive definite, radial functions with respect to the$\ell ^{p}$length on the free real line with infinite generators for$0 > p \leq 2$. We obtain a Lévy-Khintchine formula for length-radial conditionally negative functions as well.

    

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4. 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$\epsilon > 0$there exists$N > 0$such that, if$|G| \ge N$and$S, T$are normal subsets of$G$with at least$\epsilon |G|$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$\mathrm {PSL}_n(q)$where$q$is fixed and$n\to \infty$. However, in the case$S=T$and$G$alternating this holds with an explicit bound on$N$in terms of$\epsilon$.

   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$g \in G$,$P_{w_1(G),w_2(G)}(g)$denotes the probability that$g_1g_2 = g$where$g_i \in w_i(G)$are chosen uniformly and independently, then, as$|G| \to \infty$, the distribution$P_{w_1(G),w_2(G)}$tends to the uniform distribution on$G$with respect to the$L^{\infty }$norm.

    

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5. Cohomology of line bundles on the incidence correspondence

   https://doi.org/10.1090/btran/173  

   Gao, Zhao ; Raicu, Claudiu ( January 2024 , Transactions of the American Mathematical Society, Series B)

   For a finite dimensional vector space$V$of dimension$n$, we consider the incidence correspondence (or partial flag variety)$X\subset \mathbb {P}V \times \mathbb {P}V^{\vee }$, parametrizing pairs consisting of a point and a hyperplane containing it. We completely characterize the vanishing and non-vanishing behavior of the cohomology groups of line bundles on$X$in characteristic$p>0$. If$n=3$then$X$is the full flag variety of$V$, and the characterization is contained in the thesis of Griffith from the 70s. In characteristic$0$, the cohomology groups are described for all$V$by the Borel–Weil–Bott theorem. Our strategy is to recast the problem in terms of computing cohomology of (twists of) divided powers of the cotangent sheaf on projective space, which we then study using natural truncations induced by Frobenius, along with careful estimates of Castelnuovo–Mumford regularity. When$n=3$, we recover the recursive description of characters from recent work of Linyuan Liu, while for general$n$we give character formulas for the cohomology of a restricted collection of line bundles. Our results suggest truncated Schur functions as the natural building blocks for the cohomology characters.

    

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