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1. The Vlasov–Poisson–Landau system in the weakly collisional regime

   https://doi.org/10.1090/jams/1014  

   Chaturvedi, Sanchit; Luk, Jonathan; Nguyen, Toan (January 2023, Journal of the American Mathematical Society)

   Consider the Vlasov–Poisson–Landau system with Coulomb potential in the weakly collisional regime on a $3$-torus, i.e. $\begin{array}{c}{\mathrm{\partial <\#comment/>}}_{t}F(t,x,v)+v_i{\mathrm{\partial <\#comment/>}}_{x_i}F(t,x,v)+E_i(t,x){\mathrm{\partial <\#comment/>}}_{v_i}F(t,x,v)=\mathrm{\nu <\#comment/>}Q(F,F)(t,x,v),\\ E(t,x)=\mathrm{\nabla <\#comment/>}{\mathrm{\Delta <\#comment/>}}^{-<\#comment/>1}\left({\int <\#comment/>}_{{\mathbb{R}}^3}F\right(t,x,v)\mathrm{d}v-<\#comment/>{\int <\#comment/>-<\#comment/>}_{{\mathbb{T}}^3}{\int <\#comment/>}_{{\mathbb{R}}^3}F(t,x,v\left)\mathrm{d}v\mathrm{d}x\right),\end{array}$with $\mathrm{\nu <\#comment/>}\ll <\#comment/>1$. We prove that for $\mathrm{ϵ<\#comment/>}>0$sufficiently small (but independent of $\mathrm{\nu <\#comment/>}$), initial data which are $O\left(\mathrm{ϵ<\#comment/>}{\mathrm{\nu <\#comment/>}}^{1/3}\right)$-Sobolev space perturbations from the global Maxwellians lead to global-in-time solutions which converge to the global Maxwellians as $t\to <\#comment/>\mathrm{\infty <\#comment/>}$. The solutions exhibit uniform-in- $\mathrm{\nu <\#comment/>}$Landau damping and enhanced dissipation. Our main result is analogous to an earlier result of Bedrossian for the Vlasov–Poisson–Fokker–Planck equation with the same threshold. However, unlike in the Fokker–Planck case, the linear operator cannot be inverted explicitly due to the complexity of the Landau collision operator. For this reason, we develop an energy-based framework, which combines Guo’s weighted energy method with the hypocoercive energy method and the commuting vector field method. The proof also relies on pointwise resolvent estimates for the linearized density equation. 

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2. 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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3. 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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4. Spectroscopic study of the 4 f ⁷ 6 s ^{2 8} S 7/2∘−4 f ⁷ ( ⁸ S ^∘ )6 s 6 p ( ¹ P ^∘ ) ⁸ P _9/2 transition in neutral europium-151 and europium-153: absolute frequency and hyperfine structure

   https://doi.org/10.1364/JOSAB.467968  

   Herd, M_T; Maruko, C.; Herzog, M_M; Brand, A.; Cannon, G.; Duah, B.; Hollin, N.; Karani, T.; Wallace, A.; Whitmore, M.; et al (September 2022, Journal of the Optical Society of America B)

   We report on spectroscopic measurements on the $4f^{7}6s^2{}^8{S}_{7/2}^{\circ <\#comment/>}\to <\#comment/>4f^7{(}^8{S}^{\circ <\#comment/>}\left)6s6p{(}^1{P}^{\circ <\#comment/>}\right){}^8{P}_{9/2}$transition in neutral europium-151 and europium-153 at 459.4 nm. The center of gravity frequencies for the 151 and 153 isotopes, reported for the first time in this paper, to our knowledge, were found to be 652,389,757.16(34) MHz and 652,386,593.2(5) MHz, respectively. The hyperfine coefficients for the $6s6p{(}^1{P}^{\circ <\#comment/>}){}^8{P}_{9/2}$state were found to be $\mathrm{A}\left(151\right)=-<\#comment/>228.84\left(2\right)\mathrm{M}\mathrm{H}\mathrm{z}$, $\mathrm{B}\left(151\right)=226.9\left(5\right)\mathrm{M}\mathrm{H}\mathrm{z}$and $\mathrm{A}\left(153\right)=-<\#comment/>101.87\left(6\right)\mathrm{M}\mathrm{H}\mathrm{z}$, $\mathrm{B}\left(153\right)=575.4\left(1.5\right)\mathrm{M}\mathrm{H}\mathrm{z}$, which all agree with previously published results except for A(153), which shows a small discrepancy. The isotope shift is found to be 3163.8(6) MHz, which also has a discrepancy with previously published results. 

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5. Colength one deformation rings

   https://doi.org/10.1090/tran/9191  

   Le, Daniel; Le_Hung, Bao; Morra, Stefano; Park, Chol; Qian, Zicheng (May 2024, Transactions of the American Mathematical Society)

   Let $K/{\mathbb{Q}}_p$be a finite unramified extension, $\stackrel{¯<\#comment/>}{\mathrm{\rho <\#comment/>}}:\mathrm{G}\mathrm{a}\mathrm{l}\left({\stackrel{¯<\#comment/>}{\mathbb{Q}}}_p/K\right)\to <\#comment/>{\mathrm{G}\mathrm{L}}_n\left({\stackrel{¯<\#comment/>}{\mathbb{F}}}_p\right)$a continuous representation, and $\mathrm{\tau <\#comment/>}$a tame inertial type of dimension $n$. We explicitly determine, under mild regularity conditions on $\mathrm{\tau <\#comment/>}$, the potentially crystalline deformation ring $R_{\stackrel{¯<\#comment/>}{\mathrm{\rho <\#comment/>}}}^{\mathrm{\eta <\#comment/>},\mathrm{\tau <\#comment/>}}$in parallel Hodge–Tate weights $\mathrm{\eta <\#comment/>}=(n-<\#comment/>1,\cdots <\#comment/>,1,0)$and inertial type $\mathrm{\tau <\#comment/>}$when theshapeof $\stackrel{¯<\#comment/>}{\mathrm{\rho <\#comment/>}}$with respect to $\mathrm{\tau <\#comment/>}$has colength at most one. This has application to the modularity of a class of shadow weights in the weight part of Serre’s conjecture. Along the way we make unconditional the local-global compatibility results of Park and Qian [Mém. Soc. Math. Fr. (N.S.) 173 (2022), pp. vi+150]. 

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