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1. 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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2. 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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3. On higher regularity of Stokes systems with piecewise Hölder continuous coefficients

   https://doi.org/10.1090/tran/9165  

   Dong, Hongjie; Li, Haigang; Xu, Longjuan (September 2024, Transactions of the American Mathematical Society)

   In this paper, we consider higher regularity of a weak solution $(\mathbf{u},p)$to stationary Stokes systems with variable coefficients. Under the assumptions that coefficients and data are piecewise $C^{s,\mathrm{\delta <\#comment/>}}$in a bounded domain consisting of a finite number of subdomains with interfacial boundaries in $C^{s+1,\mathrm{\mu <\#comment/>}}$, where $s$is a positive integer, $\mathrm{\delta <\#comment/>}\in <\#comment/>(0,1)$, and $\mathrm{\mu <\#comment/>}\in <\#comment/>(0,1]$, we show that $D\mathbf{u}$and $p$are piecewise $C^{s,{\mathrm{\delta <\#comment/>}}_{\mathrm{\mu <\#comment/>}}}$, where ${\mathrm{\delta <\#comment/>}}_{\mathrm{\mu <\#comment/>}}=min\{\frac{1}{2},\mathrm{\mu <\#comment/>},\mathrm{\delta <\#comment/>}\}$. Our result is new even in the 2D case with piecewise constant coefficients. 

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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. 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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