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1. Violent Nonlinear Collapse in the Interior of Charged Hairy Black Holes

   https://doi.org/10.1007/s00205-024-02038-z  

   Van_de_Moortel, Maxime (October 2024, Archive for Rational Mechanics and Analysis)

   Abstract We construct a new one-parameter family, indexed by$$\epsilon $$ $ϵ$, of two-ended, spatially-homogeneous black hole interiors solving the Einstein–Maxwell–Klein–Gordon equations with a (possibly zero) cosmological constant$$\Lambda $$ $\Lambda$and bifurcating off a Reissner–Nordström-(dS/AdS) interior ($$\epsilon =0$$ $ϵ=0$). For all small$$\epsilon \ne 0$$ $ϵ\ne 0$, we prove that, although the black hole is charged, its terminal boundary is an everywhere-spacelikeKasner singularity foliated by spheres of zero radiusr. Moreover, smaller perturbations (i.e. smaller$$|\epsilon |$$ $\left|ϵ\right|$) aremore singular than larger ones, in the sense that the Hawking mass and the curvature blow up following a power law of the form$$r^{-O(\epsilon ^{-2})}$$ $r^{-O\left({ϵ}^{-2}\right)}$at the singularity$$\{r=0\}$$ $\{r=0\}$. This unusual property originates from a dynamical phenomenon—violent nonlinear collapse—caused by the almost formation of a Cauchy horizon to the past of the spacelike singularity$$\{r=0\}$$ $\{r=0\}$. This phenomenon was previously described numerically in the physics literature and referred to as “the collapse of the Einstein–Rosen bridge”. While we cover all values of$$\Lambda \in \mathbb {R}$$ $\Lambda \in R$, the case$$\Lambda <0$$ $\Lambda <0$is of particular significance to the AdS/CFT correspondence. Our result can also be viewed in general as a first step towards the understanding of the interior of hairy black holes. 

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2. Regularity of area minimizing currents mod p

   https://doi.org/10.1007/s00039-020-00546-0  

   De Lellis, Camillo; Hirsch, Jonas; Marchese, Andrea; Stuvard, Salvatore (October 2020, Geometric and Functional Analysis)

   Abstract We establish a first general partial regularity theorem for area minimizing currents$${\mathrm{mod}}(p)$$ $\mathrm{mod}\left(p\right)$, for everyp, in any dimension and codimension. More precisely, we prove that the Hausdorff dimension of the interior singular set of anm-dimensional area minimizing current$${\mathrm{mod}}(p)$$ $\mathrm{mod}\left(p\right)$cannot be larger than$$m-1$$ $m-1$. Additionally, we show that, whenpis odd, the interior singular set is$$(m-1)$$ $(m-1)$-rectifiable with locally finite$$(m-1)$$ $(m-1)$-dimensional measure. 

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3. Spherical convex hull of random points on a wedge

   https://doi.org/10.1007/s00208-023-02704-9  

   Besau, Florian; Gusakova, Anna; Reitzner, Matthias; Schütt, Carsten; Thäle, Christoph; Werner, Elisabeth_M (August 2023, Mathematische Annalen)

   Abstract Consider two half-spaces$$H_1^+$$ $H_1^{+}$and$$H_2^+$$ $H_2^{+}$in$${\mathbb {R}}^{d+1}$$ $R^{d+1}$whose bounding hyperplanes$$H_1$$ $H_1$and$$H_2$$ $H_2$are orthogonal and pass through the origin. The intersection$${\mathbb {S}}_{2,+}^d:={\mathbb {S}}^d\cap H_1^+\cap H_2^+$$ $S_{2,+}^d:=S^d\cap H_1^{+}\cap H_2^{+}$is a spherical convex subset of thed-dimensional unit sphere$${\mathbb {S}}^d$$ $S^d$, which contains a great subsphere of dimension$$d-2$$ $d-2$and is called a spherical wedge. Choosenindependent random points uniformly at random on$${\mathbb {S}}_{2,+}^d$$ $S_{2,+}^d$and consider the expected facet number of the spherical convex hull of these points. It is shown that, up to terms of lower order, this expectation grows like a constant multiple of$$\log n$$ $logn$. A similar behaviour is obtained for the expected facet number of a homogeneous Poisson point process on$${\mathbb {S}}_{2,+}^d$$ $S_{2,+}^d$. The result is compared to the corresponding behaviour of classical Euclidean random polytopes and of spherical random polytopes on a half-sphere. 

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4. Introducing isodynamic points for binary forms and their ratios

   https://doi.org/10.1007/s40627-022-00112-4  

   Hägg, Christian; Shapiro, Boris; Shapiro, Michael (January 2023, Complex Analysis and its Synergies)

   Abstract The isodynamic points of a plane triangle are known to be the only pair of its centers invariant under the action of the Möbius group$${\mathcal {M}}$$ $M$on the set of triangles, Kimberling (Encyclopedia of Triangle Centers,http://faculty.evansville.edu/ck6/encyclopedia). Generalizing this classical result, we introduce below theisodynamicmap associating to a univariate polynomial of degree$$d\ge 3$$ $d\ge 3$with at most double roots a polynomial of degree (at most)$$2d-4$$ $2d-4$such that this map commutes with the action of the Möbius group$${\mathcal {M}}$$ $M$on the zero loci of the initial polynomial and its image. The roots of the image polynomial will be called theisodynamic pointsof the preimage polynomial. Our construction naturally extends from univariate polynomials to binary forms and further to their ratios. 

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5. Generalized Parking Function Polytopes

   https://doi.org/10.1007/s00026-023-00671-1  

   Hanada, Mitsuki; Lentfer, John; Vindas-Meléndez, Andrés_R (November 2023, Annals of Combinatorics)

   Abstract A classical parking function of lengthnis a list of positive integers$$(a_1, a_2, \ldots , a_n)$$ $(a_1,a_2,\dots ,a_n)$whose nondecreasing rearrangement$$b_1 \le b_2 \le \cdots \le b_n$$ $b_1\le b_2\le \cdots \le b_n$satisfies$$b_i \le i$$ $b_i\le i$. The convex hull of all parking functions of lengthnis ann-dimensional polytope in$${\mathbb {R}}^n$$ $R^n$, which we refer to as the classical parking function polytope. Its geometric properties have been explored in Amanbayeva and Wang (Enumer Combin Appl 2(2):Paper No. S2R10, 10, 2022) in response to a question posed by Stanley (Amer Math Mon 127(6):563–571, 2020). We generalize this family of polytopes by studying the geometric properties of the convex hull of$${\textbf{x}}$$ $x$-parking functions for$${\textbf{x}}=(a,b,\dots ,b)$$ $x=(a,b,\cdots ,b)$, which we refer to as$${\textbf{x}}$$ $x$-parking function polytopes. We explore connections between these$${\textbf{x}}$$ $x$-parking function polytopes, the Pitman–Stanley polytope, and the partial permutahedra of Heuer and Striker (SIAM J Discrete Math 36(4):2863–2888, 2022). In particular, we establish a closed-form expression for the volume of$${\textbf{x}}$$ $x$-parking function polytopes. This allows us to answer a conjecture of Behrend et al. (2022) and also obtain a new closed-form expression for the volume of the convex hull of classical parking functions as a corollary. 

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