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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. 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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3. Rigidity of nonpositively curved manifolds with convex boundary

   https://doi.org/10.1090/proc/16475  

   Ghomi, Mohammad; Spruck, Joel (November 2023, Proceedings of the American Mathematical Society)

   We show that a compact Riemannian $3$-manifold $M$with strictly convex simply connected boundary and sectional curvature $K\le <\#comment/>a\le <\#comment/>0$is isometric to a convex domain in a complete simply connected space of constant curvature $a$, provided that $K\equiv <\#comment/>a$on planes tangent to the boundary of $M$. This yields a characterization of strictly convex surfaces with minimal total curvature in Cartan-Hadamard $3$-manifolds, and extends some rigidity results of Greene-Wu, Gromov, and Schroeder-Strake. Our proof is based on a recent comparison formula for total curvature of Riemannian hypersurfaces, which also yields some dual results for $K\ge <\#comment/>a\ge <\#comment/>0$. 

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4. Super-polynomial accuracy of one dimensional randomized nets using the median of means

   https://doi.org/10.1090/mcom/3791  

   Pan, Zexin; Owen, Art (March 2023, Mathematics of Computation)

   Let $f$be analytic on $[0,1]$with $|f^{\left(k\right)}\left(1/2\right)|⩽<\#comment/>A{\mathrm{\alpha <\#comment/>}}^{k}k!$for some constants $A$and $\mathrm{\alpha <\#comment/>}>2$and all $k⩾<\#comment/>1$. We show that the median estimate of $\mathrm{\mu <\#comment/>}={\int <\#comment/>}_0^{1}f\left(x\right)\mathrm{d}x$under random linear scrambling with $n=2^m$points converges at the rate $O\left(n^{-<\#comment/>c\log <\#comment/>\left(n\right)}\right)$for any $c>3\log <\#comment/>\left(2\right)/{\mathrm{\pi <\#comment/>}}^2\approx <\#comment/>0.21$. We also get a super-polynomial convergence rate for the sample median of $2k-<\#comment/>1$random linearly scrambled estimates, when $k/m$is bounded away from zero. When $f$has a $p$’th derivative that satisfies a $\mathrm{\lambda <\#comment/>}$-Hölder condition then the median of means has error $O\left(n^{-<\#comment/>(p+\mathrm{\lambda <\#comment/>})+\mathrm{ϵ<\#comment/>}}\right)$for any $\mathrm{ϵ<\#comment/>}>0$, if $k\to <\#comment/>\mathrm{\infty <\#comment/>}$as $m\to <\#comment/>\mathrm{\infty <\#comment/>}$. The proof techniques use methods from analytic combinatorics that have not previously been applied to quasi-Monte Carlo methods, most notably an asymptotic expression from Hardy and Ramanujan on the number of partitions of a natural number. 

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5. Infinite sumsets in sets with positive density

   https://doi.org/10.1090/jams/1030  

   Kra, Bryna; Moreira, Joel; Richter, Florian; Robertson, Donald (August 2023, Journal of the American Mathematical Society)

   Motivated by questions asked by Erdős, we prove that any set $A\subset <\#comment/>\mathbb{N}$with positive upper density contains, for any $k\in <\#comment/>\mathbb{N}$, a sumset $B_1+\cdots <\#comment/>+B_k$, where $B_1$, …, $B_k\subset <\#comment/>\mathbb{N}$are infinite. Our proof uses ergodic theory and relies on structural results for measure preserving systems. Our techniques are new, even for the previously known case of $k=2$. 

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