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1. On regularity of \overline∂-solutions on 𝑎_{𝑞} domains with 𝐶² boundary in complex manifolds

   https://doi.org/10.1090/tran/9315  

   Gong, Xianghong (March 2025, Transactions of the American Mathematical Society)

   We study regularity of solutions $u$to $\stackrel{¯<\#comment/>}{\mathrm{\partial <\#comment/>}}u=f$on a relatively compact $C^2$domain $D$in a complex manifold of dimension $n$, where $f$is a $(0,q)$form. Assume that there are either $(q+1)$negative or $(n-<\#comment/>q)$positive Levi eigenvalues at each point of boundary $\mathrm{\partial <\#comment/>}D$. Under the necessary condition that a locally $L^2$solution exists on the domain, we show the existence of the solutions on the closure of the domain that gain $1/2$derivative when $q=1$and $f$is in the Hölder–Zygmund space ${\mathrm{\Lambda <\#comment/>}}^r\left(D\right)$with $r>1$. For $q>1$, the same regularity for the solutions is achieved when $\mathrm{\partial <\#comment/>}D$is either sufficiently smooth or of $(n-<\#comment/>q)$positive Levi eigenvalues everywhere on $\mathrm{\partial <\#comment/>}D$. 

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2. Compatible ideals in ℚ-Gorenstein rings

   https://doi.org/10.1090/proc/16331  

   Polstra, Thomas; Schwede, Karl (October 2023, Proceedings of the American Mathematical Society)

   Suppose $R$is a $F$-finite and $F$-pure $\mathbb{Q}$-Gorenstein local ring of prime characteristic $p>0$. We show that an ideal $I\subseteq <\#comment/>R$is uniformly compatible ideal (with all $p^{-<\#comment/>e}$-linear maps) if and only if exists a module finite ring map $R\to <\#comment/>S$such that the ideal $I$is the sum of images of all $R$-linear maps $S\to <\#comment/>R$. In other words, the set of uniformly compatible ideals is exactly the set of trace ideals of finite ring maps. 

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3. A single-point Reshetnyak’s theorem

   https://doi.org/10.1090/tran/9415  

   Kangasniemi, Ilmari; Onninen, Jani (May 2025, Transactions of the American Mathematical Society)

   We prove a single-value version of Reshetnyak’s theorem. Namely, if a non-constant map $f\in W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1,n}(\mathrm{\Omega },{\mathbb{R}}^n)$ from a domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$ satisfies the estimate $\left|Df\right(x){|}^n\le K{J}_f(x)+\mathrm{\Sigma }(x\left)\right|f\left(x\right)-y_0{|}^n$ at almost every $x\in \mathrm{\Omega }$ for some $K\ge 1$, $y_0\in {\mathbb{R}}^n$ and $\mathrm{\Sigma }\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1+\epsilon }\left(\mathrm{\Omega }\right)$, then $f^{-1}\left\{y_0\right\}$ is discrete, the local index $i(x,f)$ is positive in $f^{-1}\left\{y_0\right\}$, and every neighborhood of a point of $f^{-1}\left\{y_0\right\}$ is mapped to a neighborhood of $y_0$. Assuming this estimate for a fixed $K$ at every $y_0\in {\mathbb{R}}^n$ is equivalent to assuming that the map $f$ is $K$-quasiregular, even if the choice of $\mathrm{\Sigma }$ is different for each $y_0$. Since the estimate also yields a single-value Liouville theorem, it hence appears to be a good pointwise definition of $K$-quasiregularity. As a corollary of our single-value Reshetnyak’s theorem, we obtain a higher-dimensional version of the argument principle that played a key part in the solution to the Calderón problem. 

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4. A modular construction of unramified 𝑝-extensions of ℚ(ℕ^{1/𝕡})

   https://doi.org/10.1090/bproc/141  

   Lang, Jaclyn; Wake, Preston (October 2022, Proceedings of the American Mathematical Society, Series B)

   We show that for primes $N,p\ge <\#comment/>5$with $N\equiv <\#comment/>-<\#comment/>1modp$, the class number of $\mathbb{Q}\left(N^{1/p}\right)$is divisible by $p$. Our methods are via congruences between Eisenstein series and cusp forms. In particular, we show that when $N\equiv <\#comment/>-<\#comment/>1modp$, there is always a cusp form of weight $2$and level ${\mathrm{\Gamma <\#comment/>}}_0\left(N^2\right)$whose $\mathrm{\ell <\#comment/>}$th Fourier coefficient is congruent to $\mathrm{\ell <\#comment/>}+1$modulo a prime above $p$, for all primes $\mathrm{\ell <\#comment/>}$. We use the Galois representation of such a cusp form to explicitly construct an unramified degree- $p$extension of $\mathbb{Q}\left(N^{1/p}\right)$. 

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5. Nuclear dimension of graph 𝐶*-algebras with Condition (K)

   https://doi.org/10.1090/proc/16930  

   Faurot, Gregory; Schafhauser, Christopher (October 2024, Proceedings of the American Mathematical Society)

   We prove that for any countable directed graph $E$with Condition (K), the associated graph $C^{\ast <\#comment/>}$-algebra $C^{\ast <\#comment/>}\left(E\right)$has nuclear dimension at most $2$. Furthermore, we provide a sufficient condition producing an upper bound of $1$. 

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