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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. Regularity, singularities and ℎ-vector of graded algebras

   https://doi.org/10.1090/tran/9089  

   Dao, Hailong; Ma, Linquan; Varbaro, Matteo (January 2024, Transactions of the American Mathematical Society)

   Let $R$be a standard graded algebra over a field. We investigate how the singularities of $\mathrm{Spec}<\#comment/>R$or $\mathrm{Proj}<\#comment/>R$affect the $h$-vector of $R$, which is the coefficient of the numerator of its Hilbert series. The most concrete consequence of our work asserts that if $R$satisfies Serre’s condition $\left(S_r\right)$and has reasonable singularities (Du Bois on the punctured spectrum or $F$-pure), then $h_0$, …, $h_r\ge <\#comment/>0$. Furthermore the multiplicity of $R$is at least $h_0+h_1+\cdots <\#comment/>+h_{r-<\#comment/>1}$. We also prove that equality in many cases forces $R$to be Cohen-Macaulay. The main technical tools are sharp bounds on regularity of certain $\mathrm{Ext}$modules, which can be viewed as Kodaira-type vanishing statements for Du Bois and $F$-pure singularities. Many corollaries are deduced, for instance that nice singularities of small codimension must be Cohen-Macaulay. Our results build on and extend previous work by de Fernex-Ein, Eisenbud-Goto, Huneke-Smith, Murai-Terai and others. 

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4. An optimal approximation problem for free polynomials

   https://doi.org/10.1090/proc/16474  

   Arora, Palak; Augat, Meric; Jury, Michael; Sargent, Meredith (November 2023, Proceedings of the American Mathematical Society)

   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 $c_n≔\Vert <\#comment/>p_{n}f-<\#comment/>1{\Vert <\#comment/>}^2$. (Here the norm is the ${\mathrm{\ell <\#comment/>}}^2$norm on coefficients.) We show that $c_n\to <\#comment/>0$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. 

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5. On flat manifold bundles and the connectivity of Haefliger’s classifying spaces

   https://doi.org/10.1090/proc/16941  

   Nariman, Sam (November 2024, Proceedings of the American Mathematical Society)

   We investigate a conjecture due to Haefliger and Thurston in the context of foliated manifold bundles. In this context, Haefliger-Thurston’s conjecture predicts that every $M$-bundle over a manifold $B$where $\dim <\#comment/>\left(B\right)\le <\#comment/>\dim <\#comment/>\left(M\right)$is cobordant to a flat $M$-bundle. In particular, we study the bordism class of flat $M$-bundles over low dimensional manifolds, comparing a finite dimensional Lie group $G$with ${\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}}_0\left(G\right)$. 

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