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1. 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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2. Quasi-monolithic heterodyne laser interferometer for inertial sensing

   https://doi.org/10.1364/OL.473476  

   Zhang, Yanqi; Guzman, Felipe (January 2022, Optics Letters)

   We present a compact heterodyne laser interferometer developed for high-sensitivity displacement sensing applications. This interferometer consists of customized prisms and wave plates assembled as a quasi-monolithic unit to realize a miniaturized system. The interferometer design adopts a common-mode rejection scheme to provide a high rejection ratio to common environmental noise. Experimental tests in vacuum show a displacement sensitivity level of $11\mathrm{p}\mathrm{m}/\sqrt{\mathrm{H}\mathrm{z}}$at $100\mathrm{m}\mathrm{H}\mathrm{z}$and as low as $0.6\mathrm{p}\mathrm{m}/\sqrt{\mathrm{H}\mathrm{z}}$above $1\mathrm{p}\mathrm{m}$. The prototype unit is $20\mathrm{m}\mathrm{m}\times <\#comment/>20\mathrm{m}\mathrm{m}\times <\#comment/>10\mathrm{m}\mathrm{m}$in size and weighs $4.5\mathrm{g}$, allowing subsequent integration in compact systems. 

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3. 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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4. 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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5. 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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