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1. On the maximal spectral type of nilsystems

   https://doi.org/10.1090/bproc/229  

   Ackelsberg, Ethan ; Richter, Florian ; Shalom, Or ( September 2024 , Proceedings of the American Mathematical Society, Series B)

   Let$(G/\Gamma ,R_a)$be an ergodic$k$-step nilsystem for$k\geq 2$. We adapt an argument of Parry [Topology 9 (1970), pp. 217–224] to show that$L^2(G/\Gamma )$decomposes as a sum of a subspace with discrete spectrum and a subspace of Lebesgue spectrum with infinite multiplicity. In particular, we generalize a result previously established by Host–Kra–Maass [J. Anal. Math.124(2014), pp. 261–295] for$2$-step nilsystems and a result by Stepin [Uspehi Mat. Nauk24(1969), pp. 241–242] for nilsystems$G/\Gamma$with connected, simply connected$G$.

    

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2. The structure of arbitrary Conze–Lesigne systems

   https://doi.org/10.1090/cams/31  

   Jamneshan, Asgar ; Shalom, Or ; Tao, Terence ( February 2024 , Communications of the American Mathematical Society)

   Let$\Gamma$be a countable abelian group. An (abstract)$\Gamma$-system$\mathrm {X}$- that is, an (abstract) probability space equipped with an (abstract) probability-preserving action of$\Gamma$- is said to be aConze–Lesigne systemif it is equal to its second Host–Kra–Ziegler factor$\mathrm {Z}^2(\mathrm {X})$. The main result of this paper is a structural description of such Conze–Lesigne systems for arbitrary countable abelian$\Gamma$, namely that they are the inverse limit of translational systems$G_n/\Lambda _n$arising from locally compact nilpotent groups$G_n$of nilpotency class$2$, quotiented by a lattice$\Lambda _n$. Results of this type were previously known when$\Gamma$was finitely generated, or the product of cyclic groups of prime order. In a companion paper, two of us will apply this structure theorem to obtain an inverse theorem for the Gowers$U^3(G)$norm for arbitrary finite abelian groups$G$.

    

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3. 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$\{0,1,\dots , n\}^2$has$L_1$-distortion bounded below by a constant multiple of$\sqrt {\log n}$. We provide a new “dimensionality” interpretation of Kislyakov’s argument, showing that if$\{G_n\}_{n=1}^\infty$is a sequence of graphs whose isoperimetric dimension and Lipschitz-spectral dimension equal a common number$\delta \in [2,\infty )$, then the 1-Wasserstein metric over$G_n$has$L_1$-distortion bounded below by a constant multiple of$(\log |G_n|)^{\frac {1}{\delta }}$. We proceed to compute these dimensions for$\oslash$-powers of certain graphs. In particular, we get that the sequence of diamond graphs$\{\mathsf {D}_n\}_{n=1}^\infty$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$\sqrt {\log | \mathsf {D}_n|}$. 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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4. Higher semiadditive Grothendieck-Witt theory and the 𝐾(1)-local sphere

   https://doi.org/10.1090/cams/17  

   Carmeli, Shachar ; Yuan, Allen ( March 2023 , Communications of the American Mathematical Society)

   We develop a higher semiadditive version of Grothendieck-Witt theory. We then apply the theory in the case of a finite field to study the higher semiadditive structure of the$K(1)$-local sphere$\mathbb {S}_{K(1)}$at the prime$2$, in particular realizing the non-$2$-adic rational element$1+\varepsilon \in \pi _0\mathbb {S}_{K(1)}$as a “semiadditive cardinality.” As a further application, we compute and clarify certain power operations in$\pi _0\mathbb {S}_{K(1)}$.

    

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5. Uniform Lech’s inequality

   https://doi.org/10.1090/proc/16304  

   Ma, Linquan ; Smirnov, Ilya ( February 2023 , Proceedings of the American Mathematical Society)

   Let$(R,\mathfrak {m})$be a Noetherian local ring of dimension$d\geq 2$. We prove that if$e(\widehat {R}_{red})>1$, then the classical Lech’s inequality can be improved uniformly for all$\mathfrak {m}$-primary ideals, that is, there exists$\varepsilon >0$such that$e(I)\leq d!(e(R)-\varepsilon )\ell (R/I)$for all$\mathfrak {m}$-primary ideals$I\subseteq R$. This answers a question raised by Huneke, Ma, Quy, and Smirnov [Adv. Math. 372 (2020), pp. 107296, 33]. We also obtain partial results towards improvements of Lech’s inequality when we fix the number of generators of$I$.

    

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