Suppose
This content will become publicly available on January 16, 2025
Let
 Award ID(s):
 2302430
 NSFPAR ID:
 10521500
 Publisher / Repository:
 American Mathematical Society
 Date Published:
 Journal Name:
 Transactions of the American Mathematical Society
 ISSN:
 00029947
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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$R$ is a$F$ finite and$F$ pure$\mathbb {Q}$ Gorenstein local ring of prime characteristic$p>0$ . We show that an ideal$I\subseteq R$ is uniformly compatible ideal (with all$p^{e}$ linear maps) if and only if exists a module finite ring map$R\to S$ such that the ideal$I$ is the sum of images of all$R$ linear maps$S\to R$ . In other words, the set of uniformly compatible ideals is exactly the set of trace ideals of finite ring maps. 
Motivated by recent work on optimal approximation by polynomials in the unit disk, we consider the following noncommutative approximation problem: for a polynomial
$f$ in$d$ freely noncommuting arguments, find a free polynomial$p_n$ , of degree at most$n$ , to minimize$c_n ≔\p_nf1\^2$ . (Here the norm is the$\ell ^2$ norm on coefficients.) We show that$c_n\to 0$ if and only if$f$ is nonsingular in a certain nc domain (the row ball), and prove quantitative bounds. As an application, we obtain a new proof of the characterization of polynomials cyclic for the$d$ shift. 
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 nontrivial 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 nontrivial 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. 
For each odd integer
$n \geq 3$ , we construct a rank3 graph$\Lambda _n$ with involution$\gamma _n$ whose real$C^*$ algebra$C^*_{\scriptscriptstyle \mathbb {R}}(\Lambda _n, \gamma _n)$ is stably isomorphic to the exotic Cuntz algebra$\mathcal E_n$ . This construction is optimal, as we prove that a rank2 graph with involution$(\Lambda ,\gamma )$ can never satisfy$C^*_{\scriptscriptstyle \mathbb {R}}(\Lambda , \gamma )\sim _{ME} \mathcal E_n$ , and Boersema reached the same conclusion for rank1 graphs (directed graphs) in [Münster J. Math.10 (2017), pp. 485–521, Corollary 4.3]. Our construction relies on a rank1 graph with involution$(\Lambda , \gamma )$ whose real$C^*$ algebra$C^*_{\scriptscriptstyle \mathbb {R}}(\Lambda , \gamma )$ is stably isomorphic to the suspension$S \mathbb {R}$ . In the Appendix, we show that the$i$ fold suspension$S^i \mathbb {R}$ is stably isomorphic to a graph algebra iff$2 \leq i \leq 1$ . 
Let
$(R,\mathfrak {m})$ be a Noetherian local ring of dimension$d\geq 2$ . We prove that if$e(\widehat {R}_{red})>1$ , then the classical Lech’s inequality can be improved uniformly for all$\mathfrak {m}$ primary ideals, that is, there exists$\varepsilon >0$ such that$e(I)\leq d!(e(R)\varepsilon )\ell (R/I)$ for all$\mathfrak {m}$ primary ideals$I\subseteq R$ . This answers a question raised by Huneke, Ma, Quy, and Smirnov [Adv. Math. 372 (2020), pp. 107296, 33]. We also obtain partial results towards improvements of Lech’s inequality when we fix the number of generators of$I$ .