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1. Derived 𝑝-adic heights and the leading coefficient of the Bertolini–Darmon–Prasanna 𝑝-adic 𝐿-function

   https://doi.org/10.1090/btran/227  

   Castella, Francesc; Hsu, Chi-Yun; Kundu, Debanjana; Lee, Yu-Sheng; Liu, Zheng (May 2025, Transactions of the American Mathematical Society, Series B)

   Let $E/\mathbf{Q}$be an elliptic curve and let $p$be an odd prime of good reduction for $E$. Let $K$be an imaginary quadratic field satisfying the classical Heegner hypothesis and in which $p$splits. The goal of this paper is two-fold: (1) we formulate a $p$-adic BSD conjecture for the $p$-adic $L$-function $L_{\mathfrak{p}}^{\mathrm{B}\mathrm{D}\mathrm{P}}$introduced by Bertolini–Darmon–Prasanna [Duke Math. J. 162 (2013), pp. 1033–1148]; and (2) for an algebraic analogue $F_{\stackrel{¯<\#comment/>}{\mathfrak{p}}}^{\mathrm{B}\mathrm{D}\mathrm{P}}$of $L_{\mathfrak{p}}^{\mathrm{B}\mathrm{D}\mathrm{P}}$, we show that the “leading coefficient” part of our conjecture holds, and that the “order of vanishing” part follows from the expected “maximal non-degeneracy” of an anticyclotomic $p$-adic height. In particular, when the Iwasawa–Greenberg Main Conjecture $\left(F_{\stackrel{¯<\#comment/>}{\mathfrak{p}}}^{\mathrm{B}\mathrm{D}\mathrm{P}}\right)=\left(L_{\mathfrak{p}}^{\mathrm{B}\mathrm{D}\mathrm{P}}\right)$is known, our results determine the leading coefficient of $L_{\mathfrak{p}}^{\mathrm{B}\mathrm{D}\mathrm{P}}$at $T=0$up to a $p$-adic unit. Moreover, by adapting the approach of Burungale–Castella–Kim [Algebra Number Theory 15 (2021), pp. 1627–1653], we prove the main conjecture for supersingular primes $p$under mild hypotheses. In the $p$-ordinary case, and under some additional hypotheses, similar results were obtained by Agboola–Castella [J. Théor. Nombres Bordeaux 33 (2021), pp 629–658], but our method is new and completely independent from theirs, and apply to all good primes. 

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2. The best constant for 𝐿^{∞}-type Gagliardo-Nirenberg inequalities

   https://doi.org/10.1090/qam/1645  

   Liu, Jian-Guo; Wang, Jinhuan (June 2024, Quarterly of Applied Mathematics)

   In this paper we derive the best constant for the following $L^{\mathrm{\infty <\#comment/>}}$-type Gagliardo-Nirenberg interpolation inequality $\Vert <\#comment/>u{\Vert <\#comment/>}_{L^{\mathrm{\infty <\#comment/>}}}\le <\#comment/>C_{q,\mathrm{\infty <\#comment/>},p}\Vert <\#comment/>u{\Vert <\#comment/>}_{L^{q+1}}^{1-<\#comment/>\mathrm{\theta <\#comment/>}}\Vert <\#comment/>\mathrm{\nabla <\#comment/>}u{\Vert <\#comment/>}_{L^p}^{\mathrm{\theta <\#comment/>}},\mathrm{\theta <\#comment/>}=\frac{pd}{dp+(p-<\#comment/>d)(q+1)},$where parameters $q$and $p$satisfy the conditions $p>d\ge <\#comment/>1$, $q\ge <\#comment/>0$. The best constant $C_{q,\mathrm{\infty <\#comment/>},p}$is given by $C_{q,\mathrm{\infty <\#comment/>},p}={\mathrm{\theta <\#comment/>}}^{-<\#comment/>\frac{\mathrm{\theta <\#comment/>}}{p}}(1-<\#comment/>\mathrm{\theta <\#comment/>}{)}^{\frac{\mathrm{\theta <\#comment/>}}{p}}{M}_c^{-<\#comment/>\frac{\mathrm{\theta <\#comment/>}}{d}},M_c≔{\int <\#comment/>}_{{\mathbb{R}}^d}{u}_{c,\mathrm{\infty <\#comment/>}}^{q+1}dx,$where $u_{c,\mathrm{\infty <\#comment/>}}$is the unique radial non-increasing solution to a generalized Lane-Emden equation. The case of equality holds when $u=A{u}_{c,\mathrm{\infty <\#comment/>}}\left(\mathrm{\lambda <\#comment/>}\right(x-<\#comment/>x_0\left)\right)$for any real numbers $A$, $\mathrm{\lambda <\#comment/>}>0$and $x_0\in <\#comment/>{\mathbb{R}}^d$. In fact, the generalized Lane-Emden equation in ${\mathbb{R}}^d$contains a delta function as a source and it is a Thomas-Fermi type equation. For $q=0$or $d=1$, $u_{c,\mathrm{\infty <\#comment/>}}$have closed form solutions expressed in terms of the incomplete Beta functions. Moreover, we show that $u_{c,m}\to <\#comment/>u_{c,\mathrm{\infty <\#comment/>}}$and $C_{q,m,p}\to <\#comment/>C_{q,\mathrm{\infty <\#comment/>},p}$as $m\to <\#comment/>+\mathrm{\infty <\#comment/>}$for $d=1$, where $u_{c,m}$and $C_{q,m,p}$are the function achieving equality and the best constant of $L^m$-type Gagliardo-Nirenberg interpolation inequality, respectively. 

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3. Algebraic cobordism and a Conner–Floyd isomorphism for algebraic K-theory

   https://doi.org/10.1090/jams/1045  

   Annala, Toni; Hoyois, Marc; Iwasa, Ryomei (March 2024, Journal of the American Mathematical Society)

   We formulate and prove a Conner–Floyd isomorphism for the algebraic K-theory of arbitrary qcqs derived schemes. To that end, we study a stable $\mathrm{\infty <\#comment/>}$-category of non- ${\mathbb{A}}^1$-invariant motivic spectra, which turns out to be equivalent to the $\mathrm{\infty <\#comment/>}$-category of fundamental motivic spectra satisfying elementary blowup excision, previously introduced by the first and third authors. We prove that this $\mathrm{\infty <\#comment/>}$-category satisfies ${\mathbb{P}}^1$-homotopy invariance and weighted ${\mathbb{A}}^1$-homotopy invariance, which we use in place of ${\mathbb{A}}^1$-homotopy invariance to obtain analogues of several key results from ${\mathbb{A}}^1$-homotopy theory. These allow us in particular to define a universal oriented motivic ${\mathbb{E}}_{\mathrm{\infty <\#comment/>}}$-ring spectrum $\mathrm{M}\mathrm{G}\mathrm{L}$. We then prove that the algebraic K-theory of a qcqs derived scheme $X$can be recovered from its $\mathrm{M}\mathrm{G}\mathrm{L}$-cohomology via a Conner–Floyd isomorphism\[ ${\mathrm{M}\mathrm{G}\mathrm{L}}^{\ast <\#comment/>\ast <\#comment/>}\left(X\right){\otimes <\#comment/>}_{\mathrm{L}}\mathbb{Z}\left[{\mathrm{\beta <\#comment/>}}^{\pm <\#comment/>1}\right]\simeq <\#comment/>\mathrm{K}{}^{\ast <\#comment/>\ast <\#comment/>}\left(X\right),$\]where $\mathrm{L}$is the Lazard ring and $\mathrm{K}{}^{p,q}\left(X\right)=\mathrm{K}{}_{2q-<\#comment/>p}\left(X\right)$. Finally, we prove a Snaith theorem for the periodized version of $\mathrm{M}\mathrm{G}\mathrm{L}$. 

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4. 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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5. 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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