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1. 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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2. 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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3. Torus actions, maximality, and non-negative curvature

   https://doi.org/10.1515/crelle-2021-0035  

   Escher, Christine; Searle, Catherine (August 2021, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract Let ℳ 0 n {\mathcal{M}_{0}^{n}} be the class of closed, simply connected, non-negatively curved Riemannian n -manifolds admitting an isometric, effective, isotropy-maximal torus action. We prove that if M ∈ ℳ 0 n {M\in\mathcal{M}_{0}^{n}} , then M is equivariantly diffeomorphic to the free, linear quotient by a torus of a product of spheres of dimensions greater than or equal to 3. As a special case, we then prove the Maximal Symmetry Rank Conjecture for all M ∈ ℳ 0 n {M\in\mathcal{M}_{0}^{n}} . Finally, we showthe Maximal Symmetry Rank Conjecture for simply connected, non-negatively curved manifolds holds for dimensions less than or equal to 9 without additional assumptions on the torus action. 

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4. First order rigidity of homeomorphism groups of manifolds

   https://doi.org/10.1090/cams/47  

   Kim, Sang-hyun; Koberda, Thomas; de_la_Nuez_González, J. (May 2025, Communications of the American Mathematical Society)

   For every compact, connected manifold $M$, we prove the existence of a sentence ${\mathrm{\varphi <\#comment/>}}_M$in the language of groups such that the homeomorphism group of another compact manifold $N$satisfies ${\mathrm{\varphi <\#comment/>}}_M$if and only if $N$is homeomorphic to $M$. We prove an analogous statement for groups of homeomorphisms preserving Oxtoby–Ulam probability measures. 

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5. Type systems and maximal subgroups of Thompson’s group 𝑉

   https://doi.org/10.1090/btran/204  

   Belk, James; Bleak, Collin; Quick, Martyn; Skipper, Rachel (April 2025, Transactions of the American Mathematical Society, Series B)

   We introduce the concept of a type system  $\mathcal{P}$, that is, a partition on the set of finite words over the alphabet  $\{0,1\}$compatible with the partial action of Thompson’s group  $V$, and associate a subgroup  ${\mathrm{Stab}}_V<\#comment/>\left(\mathcal{P}\right)$of  $V$. We classify the finite simple type systems and show that the stabilizers of various simple type systems, including all finite simple type systems, are maximal subgroups of  $V$. We also find an uncountable family of pairwise nonisomorphic maximal subgroups of  $V$. These maximal subgroups occur as stabilizers of infinite simple type systems and have not been described in previous literature: specifically, they do not arise as stabilizers in $V$of finite sets of points in Cantor space. Finally, we show that two natural conditions on subgroups of $V$(both related to primitivity) are each satisfied only by $V$itself, giving new ways to recognise when a subgroup of $V$is not actually proper. 

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