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1. 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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2. The principal Floquet bundle and the dynamics of fast diffusing communities

   https://doi.org/10.1090/tran/8975  

   Lam, King-Yeung; Lou, Yuan (October 2023, Transactions of the American Mathematical Society)

   We consider, for $N\ge <\#comment/>2$, the system of $N$competing species which are ecologically identical and having distinct diffusion rates $\{D_i{\}}_{i=1}^N$, in an environment with the carrying capacity $m(x,t)$. For a generic class of $m(x,t)$that varies with space and time, we show that there is a positive number $D_{\ast <\#comment/>}$independent of $N$so that if $D_i\ge <\#comment/>D_{\ast <\#comment/>}$for all $1\le <\#comment/>i\le <\#comment/>N$, then the slowest diffusing species is able to competitively exclude all other species. In the case when the environment is temporally constant or temporally periodic, our result provides some further evidence in the affirmative direction regarding the conjecture by Dockery et al. [J. Math. Biol. 37 (1998), pp. 61–83]. The main tool is the theory of the principal Floquet bundle for linear parabolic equations. 

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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. 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. 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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