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1. 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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2. 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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3. 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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4. Equivariant prime ideals for infinite dimensional supergroups

   https://doi.org/10.1090/tran/9202  

   Laudone, Robert; Snowden, Andrew (September 2024, Transactions of the American Mathematical Society)

   Let $A$be a commutative algebra equipped with an action of a group $G$. The so-called $G$-primes of $A$are the equivariant analogs of prime ideals, and of central importance in equivariant commutative algebra. When $G$is an infinite dimensional group, these ideals can be very subtle: for instance, distinct $G$-primes can have the same radical. In previous work, the second author showed that if $G={\mathbf{G}\mathbf{L}}_{\mathrm{\infty }}$and $A$is a polynomial representation, then these pathologies disappear when $G$is replaced with the supergroup ${\mathbf{G}\mathbf{L}}_{\mathrm{\infty }|\mathrm{\infty }}$and $A$with a corresponding algebra; this leads to a geometric description of $G$-primes of $A$. In the present paper, we construct an abstract framework around this result, and apply the framework to prove analogous results for other (super)groups. We give some applications to the isomeric determinantal ideals (commonly known as “queer determinantal ideals”). 

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5. A Volume = Multiplicity formula for p-families of ideals

   https://doi.org/10.1090/proc/16451  

   Das, Sudipta (October 2023, Proceedings of the American Mathematical Society)

   In this article, we work with certain families of ideals called $p$-families in rings of prime characteristic. This family of ideals is present in the theories of tight closure, Hilbert-Kunz multiplicity, and $F$-signature. For each $p$-family of ideals, we attach an Euclidean object called $p$-body, which is analogous to the Newton Okounkov body associated with a graded family of ideals. Using the combinatorial properties of $p$-bodies and algebraic properties of the Hilbert-Kunz multiplicity, we establish a Volume = Multiplicity formula for $p$-families of ${\mathfrak{m}}_R$-primary ideals in a Noetherian local ring $R$. 

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