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1. 𝐿₁-distortion of Wasserstein metrics: A tale of two dimensions

   https://doi.org/10.1090/btran/143  

   Baudier, F.; Gartland, C.; Schlumprecht, Th. (August 2023, Transactions of the American Mathematical Society, Series B)

   By discretizing an argument of Kislyakov, Naor and Schechtman proved that the 1-Wasserstein metric over the planar grid $\{0,1,\dots <\#comment/>,n{\}}^2$has $L_1$-distortion bounded below by a constant multiple of $\sqrt{\log <\#comment/>n}$. We provide a new “dimensionality” interpretation of Kislyakov’s argument, showing that if $\{G_n{\}}_{n=1}^{\mathrm{\infty <\#comment/>}}$is a sequence of graphs whose isoperimetric dimension and Lipschitz-spectral dimension equal a common number $\mathrm{\delta <\#comment/>}\in <\#comment/>[2,\mathrm{\infty <\#comment/>})$, then the 1-Wasserstein metric over $G_n$has $L_1$-distortion bounded below by a constant multiple of $(\log <\#comment/>|G_n|{)}^{\frac{1}{\mathrm{\delta <\#comment/>}}}$. We proceed to compute these dimensions for $\oslash <\#comment/>$-powers of certain graphs. In particular, we get that the sequence of diamond graphs $\{{\mathsf{D}}_n{\}}_{n=1}^{\mathrm{\infty <\#comment/>}}$has isoperimetric dimension and Lipschitz-spectral dimension equal to 2, obtaining as a corollary that the 1-Wasserstein metric over ${\mathsf{D}}_n$has $L_1$-distortion bounded below by a constant multiple of $\sqrt{\log <\#comment/>|{\mathsf{D}}_n|}$. This answers a question of Dilworth, Kutzarova, and Ostrovskii and exhibits only the third sequence of $L_1$-embeddable graphs whose sequence of 1-Wasserstein metrics is not $L_1$-embeddable. 

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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. 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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4. Cyclic isogenies of elliptic curves over fixed quadratic fields

   https://doi.org/10.1090/mcom/3894  

   Banwait, Barinder; Najman, Filip; Padurariu, Oana (March 2024, Mathematics of Computation)

   Building on Mazur’s 1978 work on prime degree isogenies, Kenku determined in 1981 all possible cyclic isogenies of elliptic curves over $\mathbb{Q}$. Although more than 40 years have passed, the determination of cyclic isogenies of elliptic curves over a single other number field has hitherto not been realised. In this paper we develop a procedure to assist in establishing such a determination for a given quadratic field. Executing this procedure on all quadratic fields $\mathbb{Q}\left(\sqrt{d}\right)$with $|d|>{10}^4$we obtain, conditional on the Generalised Riemann Hypothesis, the determination of cyclic isogenies of elliptic curves over $19$quadratic fields, including $\mathbb{Q}\left(\sqrt{213}\right)$and $\mathbb{Q}\left(\sqrt{-<\#comment/>2289}\right)$. To make this procedure work, we determine all of the finitely many quadratic points on the modular curves $X_0\left(125\right)$and $X_0\left(169\right)$, which may be of independent interest. 

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5. 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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