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1. 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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2. Borel Vizing’s theorem for graphs of subexponential growth

   https://doi.org/10.1090/proc/17048  

   Bernshteyn, Anton; Dhawan, Abhishek (January 2025, Proceedings of the American Mathematical Society)

   We show that every Borel graph $G$of subexponential growth has a Borel proper edge-coloring with $\mathrm{\Delta <\#comment/>}\left(G\right)+1$colors. We deduce this from a stronger result, namely that an $n$-vertex (finite) graph $G$of subexponential growth can be properly edge-colored using $\mathrm{\Delta <\#comment/>}\left(G\right)+1$colors by an $O({\log }^{\ast <\#comment/>}<\#comment/>n)$-round deterministic distributed algorithm in theLOCALmodel, where the implied constants in the $O(\cdot <\#comment/>)$notation are determined by a bound on the growth rate of $G$. 

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3. Bounds on cohomological support varieties

   https://doi.org/10.1090/btran/182  

   Briggs, Benjamin; Grifo, Eloísa; Pollitz, Josh (March 2024, Transactions of the American Mathematical Society, Series B)

   Over a local ring $R$, the theory of cohomological support varieties attaches to any bounded complex $M$of finitely generated $R$-modules an algebraic variety ${\mathrm{V}}_R\left(M\right)$that encodes homological properties of $M$. We give lower bounds for the dimension of ${\mathrm{V}}_R\left(M\right)$in terms of classical invariants of $R$. In particular, when $R$is Cohen–Macaulay and not complete intersection we find that there are always varieties that cannot be realized as the cohomological support of any complex. When $M$has finite projective dimension, we also give an upper bound for $\dim <\#comment/>{\mathrm{V}}_R\left(M\right)$in terms of the dimension of the radical of the homotopy Lie algebra of $R$. This leads to an improvement of a bound due to Avramov, Buchweitz, Iyengar, and Miller on the Loewy lengths of finite free complexes, and it recovers a result of Avramov and Halperin on the homotopy Lie algebra of $R$. Finally, we completely classify the varieties that can occur as the cohomological support of a complex over a Golod ring. 

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4. Dens, nests and the Loehr-Warrington conjecture

   https://doi.org/10.1090/jams/1057  

   Blasiak, J; Haiman, M; Morse, J; Pun, A; Seelinger, G (June 2025, Journal of the American Mathematical Society)

   We prove and extend the longest-standing conjecture in ‘ $q,t$-Catalan combinatorics,’ namely, the combinatorial formula for ${\mathrm{\nabla <\#comment/>}}^m{s}_{\mathrm{\mu <\#comment/>}}$conjectured by Loehr and Warrington, where $s_{\mathrm{\mu <\#comment/>}}$is a Schur function and $\mathrm{\nabla <\#comment/>}$is an eigenoperator on Macdonald polynomials. Our approach is to establish a stronger identity of infinite series of $G{L}_l$characters involvingSchur Catalanimals; these were recently shown by the authors to represent Schur functions $s_{\mathrm{\mu <\#comment/>}}[-<\#comment/>M{X}^{m,n}]$in subalgebras $\mathrm{\Lambda <\#comment/>}\left(X^{m,n}\right)\subset <\#comment/>\mathcal{E}$isomorphic to the algebra of symmetric functions $\mathrm{\Lambda <\#comment/>}$over $\mathbb{Q}(q,t)$, where $\mathcal{E}$is the elliptic Hall algebra of Burban and Schiffmann. We establish a combinatorial formula for Schur Catalanimals as weighted sums of LLT polynomials, with terms indexed by configurations of nested lattice paths callednests, having endpoints and bounding constraints controlled by data called aden. The special case for $\mathrm{\Lambda <\#comment/>}\left(X^{m,1}\right)$proves the Loehr-Warrington conjecture, giving ${\mathrm{\nabla <\#comment/>}}^m{s}_{\mathrm{\mu <\#comment/>}}$as a weighted sum of LLT polynomials indexed by systems of nested Dyck paths. In general, for $\mathrm{\Lambda <\#comment/>}\left(X^{m,n}\right)$our formula implies a new $(m,n)$version of the Loehr-Warrington conjecture. In the case where each nest consists of a single lattice path, the nests in a den formula reduce to our previous shuffle theorem for paths under any line. Both this and the $(m,n)$Loehr-Warrington formula generalize the $(km,kn)$shuffle theorem proven by Carlsson and Mellit (for $n=1$) and Mellit. Our formula here unifies these two generalizations. 

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5. Relations between Poincaré series for quasi-complete intersection homomorphisms

   https://doi.org/10.1090/proc/17078  

   Pollitz, Josh; Şega, Liana (January 2025, Proceedings of the American Mathematical Society)

   In this article we study base change of Poincaré series along a quasi-complete intersection homomorphism $\mathrm{\phi <\#comment/>}:<\#comment/>Q\to <\#comment/>R$, where $Q$is a local ring with maximal ideal $\mathfrak{m}$. In particular, we give a precise relationship between the Poincaré series ${\mathrm{P}}_M^Q\left(t\right)$of a finitely generated $R$-module $M$to ${\mathrm{P}}_M^R\left(t\right)$when the kernel of $\mathrm{\phi <\#comment/>}$is contained in $\mathfrak{m}{\mathrm{a}\mathrm{n}\mathrm{n}}_Q\left(M\right)$. This generalizes a classical result of Shamash for complete intersection homomorphisms. Our proof goes through base change formulas for Poincaré series under the map of dg algebras $Q\to <\#comment/>E$, with $E$the Koszul complex on a minimal set of generators for the kernel of $\mathrm{\phi <\#comment/>}$. 

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