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1. 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 $d\ge <\#comment/>2$. We prove that if $e\left({\stackrel{^<\#comment/>}{R}}_{red}\right)>1$, then the classical Lech’s inequality can be improved uniformly for all $\mathfrak{m}$-primary ideals, that is, there exists $\mathrm{\epsilon <\#comment/>}>0$such that $e\left(I\right)\le <\#comment/>d!\left(e\right(R)-<\#comment/>\mathrm{\epsilon <\#comment/>})\mathrm{\ell <\#comment/>}\left(R/I\right)$for all $\mathfrak{m}$-primary ideals $I\subseteq <\#comment/>R$. 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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2. Braid group action and quasi-split affine 𝚤quantum groups I

   https://doi.org/10.1090/ert/657  

   Lu, Ming; Wang, Weiqiang; Zhang, Weinan (October 2023, Representation Theory of the American Mathematical Society)

   This is the first of our papers on quasi-split affine quantum symmetric pairs $(\stackrel{~<\#comment/>}{\mathbf{U}}\left(\stackrel{^<\#comment/>}{\mathfrak{g}}\right),{\stackrel{~<\#comment/>}{\mathbf{U}}}^{\mathrm{ı<\#comment/>}})$, focusing on the real rank one case, i.e., $\mathfrak{g}={\mathfrak{s}\mathfrak{l}}_3$equipped with a diagram involution. We construct explicitly a relative braid group action of type $A_2^{\left(2\right)}$on the affine $\mathrm{ı<\#comment/>}$quantum group ${\stackrel{~<\#comment/>}{\mathbf{U}}}^{\mathrm{ı<\#comment/>}}$. Real and imaginary root vectors for ${\stackrel{~<\#comment/>}{\mathbf{U}}}^{\mathrm{ı<\#comment/>}}$are constructed, and a Drinfeld type presentation of ${\stackrel{~<\#comment/>}{\mathbf{U}}}^{\mathrm{ı<\#comment/>}}$is then established. This provides a new basic ingredient for the Drinfeld type presentation of higher rank quasi-split affine $\mathrm{ı<\#comment/>}$quantum groups in the sequels. 

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3. Stable Big Bang formation for Einstein’s equations: The complete sub-critical regime

   https://doi.org/10.1090/jams/1015  

   Fournodavlos, Grigorios; Rodnianski, Igor; Speck, Jared (July 2023, Journal of the American Mathematical Society)

   For $(t,x)\in <\#comment/>(0,\mathrm{\infty <\#comment/>})\times <\#comment/>{\mathbb{T}}^{\mathfrak{D}}$, the generalized Kasner solutions (which we refer to as Kasner solutions for short) are a family of explicit solutions to various Einstein-matter systems that, exceptional cases aside, start out smooth but then develop a Big Bang singularity as $t\downarrow <\#comment/>0$, i.e., a singularity along an entire spacelike hypersurface, where various curvature scalars blow up monotonically. The family is parameterized by the Kasner exponents ${\stackrel{~<\#comment/>}{q}}_1,\cdots <\#comment/>,{\stackrel{~<\#comment/>}{q}}_{\mathfrak{D}}\in <\#comment/>\mathbb{R}$, which satisfy two algebraic constraints. There are heuristics in the mathematical physics literature, going back more than 50 years, suggesting that the Big Bang formation should be dynamically stable, that is, stable under perturbations of the Kasner initial data, given say at $\{t=1\}$, as long as the exponents are “sub-critical” in the following sense: $\underset{\begin{array}{c}I,J,B=1,\cdots <\#comment/>,\mathfrak{D}\\ I>J\end{array}}{max}\{{\stackrel{~<\#comment/>}{q}}_I+{\stackrel{~<\#comment/>}{q}}_J-<\#comment/>{\stackrel{~<\#comment/>}{q}}_B\}>1$. Previous works have rigorously shown the dynamic stability of the Kasner Big Bang singularity under stronger assumptions: (1) the Einstein-scalar field system with $\mathfrak{D}=3$and ${\stackrel{~<\#comment/>}{q}}_1\approx <\#comment/>{\stackrel{~<\#comment/>}{q}}_2\approx <\#comment/>{\stackrel{~<\#comment/>}{q}}_3\approx <\#comment/>1/3$, which corresponds to the stability of the Friedmann–Lemaître–Robertson–Walker solution’s Big Bang or (2) the Einstein-vacuum equations for $\mathfrak{D}\ge <\#comment/>38$with $\underset{I=1,\cdots <\#comment/>,\mathfrak{D}}{max}|{\stackrel{~<\#comment/>}{q}}_I|>1/6$. In this paper, we prove that the Kasner singularity is dynamically stable forallsub-critical Kasner exponents, thereby justifying the heuristics in the literature in the full regime where stable monotonic-type curvature-blowup is expected. We treat in detail the $1+\mathfrak{D}$-dimensional Einstein-scalar field system for all $\mathfrak{D}\ge <\#comment/>3$and the $1+\mathfrak{D}$-dimensional Einstein-vacuum equations for $\mathfrak{D}\ge <\#comment/>10$; both of these systems feature non-empty sets of sub-critical Kasner solutions. Moreover, for the Einstein-vacuum equations in $1+3$dimensions, where instabilities are in general expected, we prove that all singular Kasner solutions have dynamically stable Big Bangs under polarized $U\left(1\right)$-symmetric perturbations of their initial data. Our results hold for open sets of initial data in Sobolev spaces without symmetry, apart from our work on polarized $U\left(1\right)$-symmetric solutions. Our proof relies on a new formulation of Einstein’s equations: we use a constant-mean-curvature foliation, and the unknowns are the scalar field, the lapse, the components of the spatial connection and second fundamental form relative to a Fermi–Walker transported spatial orthonormal frame, and the components of the orthonormal frame vectors with respect to a transported spatial coordinate system. In this formulation, the PDE evolution system for the structure coefficients of the orthonormal frame approximately diagonalizes in a way that sharply reveals the significance of the Kasner exponent sub-criticality condition for the dynamic stability of the flow: the condition leads to the time-integrability of many terms in the equations, at least at the low derivative levels. At the high derivative levels, the solutions that we study can be much more singular with respect to $t$, and to handle this difficulty, we use $t$-weighted high order energies, and we control non-linear error terms by exploiting monotonicity induced by the $t$-weights and interpolating between the singularity-strength of the solution’s low order and high order derivatives. Finally, we note that our formulation of Einstein’s equations highlights the quantities that might generate instabilities outside of the sub-critical regime. 

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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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