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1. Property G and the 4-genus

   https://doi.org/10.1090/btran/153  

   Ni, Yi (January 2024, Transactions of the American Mathematical Society, Series B)

   We say a null-homologous knot $K$in a $3$-manifold $Y$has Property G, if the Thurston norm and fiberedness of the complement of $K$is preserved under the zero surgery on $K$. In this paper, we will show that, if the smooth $4$-genus of $K\times <\#comment/>\left\{0\right\}$(in a certain homology class) in $(Y\times <\#comment/>[0,1\left]\right)\mathrm{\#<\#comment/>}N\stackrel{¯<\#comment/>}{\mathbb{C}{P}^2}$, where $Y$is a rational homology sphere, is smaller than the Seifert genus of $K$, then $K$has Property G. When the smooth $4$-genus is $0$, $Y$can be taken to be any closed, oriented $3$-manifold. 

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2. Simple closed curves in stable covers of surfaces

   https://doi.org/10.1090/tran/8940  

   Salter, Nick (May 2023, Transactions of the American Mathematical Society)

   Let $f:X\to <\#comment/>Y$be a regular covering of a surface $Y$of finite type with nonempty boundary, with finitely-generated (possibly infinite) deck group $G$. We give necessary and sufficient conditions for an integral homology class on $X$to admit a representative as a connected component of the preimage of a nonseparating simple closed curve on $Y$, possibly after passing to a “stabilization”, i.e. a $G$-equivariant embedding of covering spaces $X\hookrightarrow <\#comment/>X^{+}$. 

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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 $f\in <\#comment/>W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1,n}(\mathrm{\Omega <\#comment/>},{\mathbb{R}}^n)$from a domain $\mathrm{\Omega <\#comment/>}\subset <\#comment/>{\mathbb{R}}^n$satisfies the estimate $|<\#comment/>Df\left(x\right){|<\#comment/>}^n\le <\#comment/>K{J}_f\left(x\right)+\mathrm{\Sigma <\#comment/>}\left(x\right)|<\#comment/>f\left(x\right)-<\#comment/>y_0{|<\#comment/>}^n$at almost every $x\in <\#comment/>\mathrm{\Omega <\#comment/>}$for some $K\ge <\#comment/>1$, $y_0\in <\#comment/>{\mathbb{R}}^n$and $\mathrm{\Sigma <\#comment/>}\in <\#comment/>L_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1+\mathrm{\epsilon <\#comment/>}}\left(\mathrm{\Omega <\#comment/>}\right)$, then $f^{-<\#comment/>1}\left\{y_0\right\}$is discrete, the local index $i(x,f)$is positive in $f^{-<\#comment/>1}\left\{y_0\right\}$, and every neighborhood of a point of $f^{-<\#comment/>1}\left\{y_0\right\}$is mapped to a neighborhood of $y_0$. Assuming this estimate for a fixed $K$at every $y_0\in <\#comment/>{\mathbb{R}}^n$is equivalent to assuming that the map $f$is $K$-quasiregular, even if the choice of $\mathrm{\Sigma <\#comment/>}$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. 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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5. 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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