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1. 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$(t,x) \in (0,\infty )\times \mathbb {T}^{\mathfrak {D}}$, 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$t \downarrow 0$, i.e., a singularity along an entire spacelike hypersurface, where various curvature scalars blow up monotonically. The family is parameterized by the Kasner exponents$\widetilde {q}_1,\cdots ,\widetilde {q}_{\mathfrak {D}} \in \mathbb {R}$, 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$\lbrace t = 1 \rbrace$, as long as the exponents are “sub-critical” in the following sense:$\underset {\substack {I,J,B=1,\cdots , \mathfrak {D}\\ I > J}}{\max } \{\widetilde {q}_I+\widetilde {q}_J-\widetilde {q}_B\}>1$. 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$\widetilde {q}_1 \approx \widetilde {q}_2 \approx \widetilde {q}_3 \approx 1/3$, which corresponds to the stability of the Friedmann–Lemaître–Robertson–Walker solution’s Big Bang or (2) the Einstein-vacuum equations for$\mathfrak {D}\geq 38$with$\underset {I=1,\cdots ,\mathfrak {D}}{\max } |\widetilde {q}_I| > 1/6$. 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$\mathfrak {D}\geq 3$and the$1+\mathfrak {D}$-dimensional Einstein-vacuum equations for$\mathfrak {D}\geq 10$; 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$U(1)$-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$U(1)$-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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2. Super-polynomial accuracy of one dimensional randomized nets using the median of means

   https://doi.org/10.1090/mcom/3791  

   Pan, Zexin ; Owen, Art ( March 2023 , Mathematics of Computation)

   Let$f$be analytic on$[0,1]$with$|f^{(k)}(1/2)|\leqslant A\alpha ^kk!$for some constants$A$and$\alpha >2$and all$k\geqslant 1$. We show that the median estimate of$\mu =\int _0^1f(x)\,\mathrm {d} x$under random linear scrambling with$n=2^m$points converges at the rate$O(n^{-c\log (n)})$for any$c> 3\log (2)/\pi ^2\approx 0.21$. We also get a super-polynomial convergence rate for the sample median of$2k-1$random linearly scrambled estimates, when$k/m$is bounded away from zero. When$f$has a$p$’th derivative that satisfies a$\lambda$-Hölder condition then the median of means has error$O( n^{-(p+\lambda )+\epsilon })$for any$\epsilon >0$, if$k\to \infty$as$m\to \infty$. The proof techniques use methods from analytic combinatorics that have not previously been applied to quasi-Monte Carlo methods, most notably an asymptotic expression from Hardy and Ramanujan on the number of partitions of a natural number.

    

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3. 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$L^{\infty }$-type Gagliardo-Nirenberg interpolation inequality$\begin{equation*} \|u\|_{L^{\infty }}\leq C_{q,\infty ,p} \|u\|^{1-\theta }_{L^{q+1}}\|\nabla u\|^{\theta }_{L^p},\quad \theta =\frac {pd}{dp+(p-d)(q+1)}, \end{equation*}$where parameters$q$and$p$satisfy the conditions$p>d\geq 1$,$q\geq 0$. The best constant$C_{q,\infty ,p}$is given by$\begin{equation*} C_{q,\infty ,p}=\theta ^{-\frac {\theta }{p}}(1-\theta )^{\frac {\theta }{p}}M_c^{-\frac {\theta }{d}},\quad M_c≔\int _{\mathbb {R}^d}u_{c,\infty }^{q+1} dx, \end{equation*}$where$u_{c,\infty }$is the unique radial non-increasing solution to a generalized Lane-Emden equation. The case of equality holds when$u=Au_{c,\infty }(\lambda (x-x_0))$for any real numbers$A$,$\lambda >0$and$x_{0}\in \mathbb {R}^d$. 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$d=1$,$u_{c,\infty }$have closed form solutions expressed in terms of the incomplete Beta functions. Moreover, we show that$u_{c,m}\to u_{c,\infty }$and$C_{q,m,p}\to C_{q,\infty ,p}$as$m\to +\infty$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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4. An optimal approximation problem for free polynomials

   https://doi.org/10.1090/proc/16474  

   Arora, Palak ; Augat, Meric ; Jury, Michael ; Sargent, Meredith ( November 2023 , Proceedings of the American Mathematical Society)

   Motivated by recent work on optimal approximation by polynomials in the unit disk, we consider the following noncommutative approximation problem: for a polynomial$f$in$d$freely noncommuting arguments, find a free polynomial$p_n$, of degree at most$n$, to minimize$c_n ≔\|p_nf-1\|^2$. (Here the norm is the$\ell ^2$norm on coefficients.) We show that$c_n\to 0$if and only if$f$is nonsingular in a certain nc domain (the row ball), and prove quantitative bounds. As an application, we obtain a new proof of the characterization of polynomials cyclic for the$d$-shift.

    

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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$\{0,1,\dots , n\}^2$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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