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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. The embedding theorem in Hurwitz–Brill–Noether theory

   https://doi.org/10.1090/btran/232  

   Cook-Powell, Kaelin; Jensen, David; Larson, Eric; Larson, Hannah; Vogt, Isabel (January 2026, Transactions of the American Mathematical Society, Series B)

   We generalize the Embedding Theorem of Eisenbud–Harris from classical Brill–Noether theory to the setting of Hurwitz–Brill–Noether theory. More precisely, in classical Brill–Noether theory, the embedding theorem states that a general linear series of degree $d$ and rank $r$ on a general curve of genus $g$ is an embedding if $r\ge <\#comment/>3$ . If $f:<\#comment/>C\to <\#comment/>{\mathbb{P}}^1$ is a general cover of degree $k$ , and $\mathcal{L}$ is a line bundle on $C$ , recent work of the authors shows that the splitting type of $f_{\ast <\#comment/>}\mathcal{L}$ provides the appropriate generalization of the pair $(r,d)$ in classical Brill–Noether theory (see K. Cook-Powell and D. Jensen [Michigan Math. J. 71 (2022), pp. 19–45]; K. Cook-Powell and D. Jensen [Adv. Math. 398 (2022)]; E. Larson, H. Larson, and I. Vogt [Geom. Topol. 29 (2025), pp. 193–257]; and H. K. Larson [Invent. Math. 224 (2021), pp. 767–790]). In the context of Hurwitz–Brill–Noether theory, the condition $r\ge <\#comment/>3$ is no longer sufficient to guarantee that a general such linear series is an embedding. We show that the additional condition needed to guarantee that a general linear series $|\mathcal{L}|$ is an embedding is that the splitting type of $f_{\ast <\#comment/>}\mathcal{L}$ has at least three nonnegative parts. This new extra condition reflects the unique geometry of $k$ -gonal curves, which lie on scrolls in ${\mathbb{P}}^r$ . 

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4. 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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5. 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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