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1. Mapping class groups of exotic tori and actions by 𝑆𝐿_{𝑑}(𝐙)

   https://doi.org/10.1090/btran/207  

   Bustamante, Mauricio; Krannich, Manuel; Kupers, Alexander; Tshishiku, Bena (November 2024, Transactions of the American Mathematical Society, Series B)

   We determine for which exotic tori $\mathcal{T}$of dimension $d\ne <\#comment/>4$the homomorphism from the group of isotopy classes of orientation-preserving diffeomorphisms of $\mathcal{T}$to ${\mathrm{S}\mathrm{L}}_d\left(\mathbf{Z}\right)$given by the action on the first homology group is split surjective. As part of the proof we compute the mapping class group of all exotic tori $\mathcal{T}$that are obtained from the standard torus by a connected sum with an exotic sphere. Moreover, we show that any nontrivial ${\mathrm{S}\mathrm{L}}_d\left(\mathbf{Z}\right)$-action on $\mathcal{T}$agrees on homology with the standard action, up to an automorphism of ${\mathrm{S}\mathrm{L}}_d\left(\mathbf{Z}\right)$. When combined, these results in particular show that many exotic tori do not admit any nontrivial differentiable action by ${\mathrm{S}\mathrm{L}}_d\left(\mathbf{Z}\right)$. 

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2. On the Size of the Singular Set of Minimizing Harmonic Maps

   https://doi.org/10.1090/memo/1519  

   Mazowiecka, Katarzyna; Miśkiewicz, Michał; Schikorra, Armin (October 2024, Memoirs of the American Mathematical Society)

   We consider minimizing harmonic maps $u$from $\mathrm{\Omega <\#comment/>}\subset <\#comment/>{\mathbb{R}}^n$into a closed Riemannian manifold $\mathcal{N}$and prove: 1. an extension to $n\ge <\#comment/>4$of Almgren and Lieb’s linear law. That is, if the fundamental group of the target manifold $\mathcal{N}$is finite, we have\[ ${\mathcal{H}}^{n-<\#comment/>3}(\mathrm{sing}<\#comment/>u)\le <\#comment/>C{\int <\#comment/>}_{\mathrm{\partial <\#comment/>}\mathrm{\Omega <\#comment/>}}|{\mathrm{\nabla <\#comment/>}}_{T}u{|}^{n-<\#comment/>1}\mathrm{d}{\mathcal{H}}^{n-<\#comment/>1};$\]2. an extension of Hardt and Lin’s stability theorem. Namely, assuming that the target manifold is $\mathcal{N}={\mathbb{S}}^2$we obtain that the singular set of $u$is stable under small $W^{1,n-<\#comment/>1}$-perturbations of the boundary data. In dimension $n=3$both results are shown to hold with weaker hypotheses, i.e., only assuming that the trace of our map lies in the fractional space $W^{s,p}$with $s\in <\#comment/>(\frac{1}{2},1]$and $p\in <\#comment/>[2,\mathrm{\infty <\#comment/>})$satisfying $sp\ge <\#comment/>2$. We also discuss sharpness. 

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3. The 𝑚=2 amplituhedron and the hypersimplex: Signs, clusters, tilings, Eulerian numbers

   https://doi.org/10.1090/cams/23  

   Parisi, Matteo; Sherman-Bennett, Melissa; Williams, Lauren (July 2023, Communications of the American Mathematical Society)

   The hypersimplex ${\mathrm{\Delta <\#comment/>}}_{k+1,n}$is the image of the positive Grassmannian $G{r}_{k+1,n}^{\ge <\#comment/>0}$under the moment map. It is a polytope of dimension $n-<\#comment/>1$in ${\mathbb{R}}^n$. Meanwhile, the amplituhedron ${\mathcal{A}}_{n,k,2}\left(Z\right)$is the projection of the positive Grassmannian $G{r}_{k,n}^{\ge <\#comment/>0}$into the Grassmannian $G{r}_{k,k+2}$under a map $\stackrel{~<\#comment/>}{Z}$induced by a positive matrix $Z\in <\#comment/>Ma{t}_{n,k+2}^{>0}$. Introduced in the context ofscattering amplitudes, it is not a polytope, and has full dimension $2k$inside $G{r}_{k,k+2}$. Nevertheless, there seem to be remarkable connections between these two objects viaT-duality, as conjectured by Łukowski, Parisi, and Williams [Int. Math. Res. Not. (2023)]. In this paper we use ideas from oriented matroid theory, total positivity, and the geometry of the hypersimplex and positroid polytopes to obtain a deeper understanding of the amplituhedron. We show that the inequalities cutting outpositroid polytopes—images of positroid cells of $G{r}_{k+1,n}^{\ge <\#comment/>0}$under the moment map—translate into sign conditions characterizing the T-dualGrasstopes—images of positroid cells of $G{r}_{k,n}^{\ge <\#comment/>0}$under $\stackrel{~<\#comment/>}{Z}$. Moreover, we subdivide the amplituhedron intochambers, just as the hypersimplex can be subdivided into simplices, with both chambers and simplices enumerated by the Eulerian numbers. We use these properties to prove the main conjecture of Łukowski, Parisi, and Williams [Int. Math. Res. Not. (2023)]: a collection of positroid polytopes is a tiling of the hypersimplex if and only if the collection of T-dual Grasstopes is a tiling of the amplituhedron ${\mathcal{A}}_{n,k,2}\left(Z\right)$for all $Z$. Moreover, we prove Arkani-Hamed–Thomas–Trnka’s conjectural sign-flip characterization of ${\mathcal{A}}_{n,k,2}$, and Łukowski–Parisi–Spradlin–Volovich’s conjectures on $m=2$cluster adjacencyand onpositroid tilesfor ${\mathcal{A}}_{n,k,2}$(images of $2k$-dimensional positroid cells which map injectively into ${\mathcal{A}}_{n,k,2}$). Finally, we introduce new cluster structures in the amplituhedron. 

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4. Planar Algebras in Braided Tensor Categories

   https://doi.org/10.1090/memo/1392  

   Henriques, André; Penneys, David; Tener, James (February 2023, Memoirs of the American Mathematical Society)

   We generalize Jones’ planar algebras by internalising the notion to a pivotal braided tensor category $\mathcal{C}$. To formulate the notion, the planar tangles are now equipped with additional ‘anchor lines’ which connect the inner circles to the outer circle. We call the resulting notion ananchored planar algebra. If we restrict to the case when $\mathcal{C}$is the category of vector spaces, then we recover the usual notion of a planar algebra. Building on our previous work on categorified traces, we prove that there is an equivalence of categories between anchored planar algebras in $\mathcal{C}$and pivotal module tensor categories over $\mathcal{C}$equipped with a chosen self-dual generator. Even in the case of usual planar algebras, the precise formulation of this theorem, as an equivalence of categories, has not appeared in the literature. Using our theorem, we describe many examples of anchored planar algebras. 

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5. Sums of linear transformations

   https://doi.org/10.1090/tran/9433  

   Conlon, David; Lim, Jeck (July 2025, Transactions of the American Mathematical Society)

   We show that if ${\mathcal{L}}_1$and ${\mathcal{L}}_2$are linear transformations from ${\mathbb{Z}}^d$to ${\mathbb{Z}}^d$satisfying certain mild conditions, then, for any finite subset $A$of ${\mathbb{Z}}^d$, $|{\mathcal{L}}_{1}A+{\mathcal{L}}_{2}A|\ge <\#comment/>{(|det({\mathcal{L}}_1){|}^{1/d}+|det({\mathcal{L}}_2\left){|}^{1/d}\right)}^d|A|-<\#comment/>o\left(|A|\right).$This result corrects and confirms the two-summand case of a conjecture of Bukh and is best possible up to the lower-order term for certain choices of ${\mathcal{L}}_1$and ${\mathcal{L}}_2$. As an application, we prove a lower bound for $|A+\mathrm{\lambda <\#comment/>}\cdot <\#comment/>A|$when $A$is a finite set of real numbers and $\mathrm{\lambda <\#comment/>}$is an algebraic number. In particular, when $\mathrm{\lambda <\#comment/>}$is of the form $(p/q{)}^{1/d}$for some $p,q,d\in <\#comment/>\mathbb{N}$, each taken as small as possible for such a representation, we show that $|A+\mathrm{\lambda <\#comment/>}\cdot <\#comment/>A|\ge <\#comment/>(p^{1/d}+q^{1/d}{)}^d|A|-<\#comment/>o(|A|).$This is again best possible up to the lower-order term and extends a recent result of Krachun and Petrov which treated the case $\mathrm{\lambda <\#comment/>}=\sqrt{2}$. 

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