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### Vol. 64, no. 1 (2022) Special volume: Mathematical Congress of the Americas 2021

August 2022
Editors: Guillermo Cortiñas (managing editor, Universidad de Buenos Aires); Renato Iturriaga (CIMAT, Guanajuato, Mexico); Andrea Nahmod (University of Massachusetts, Amherst, USA); Jimmy Petean (CIMAT, Guanajuato, Mexico); Arturo Pianzola (University of Alberta, Canada and Universidad CAECE, Argentina); Cecilia Salgado (University of Groningen, Netherlands and Universidade Federal do Rio de Janeiro, Brazil); Soledad Torres (CIMFAV-INGEMAT, Universidad de Valparaíso, Chile); Bernardo Uribe (Universidad del Norte, Colombia); Steven Weintraub (Lehigh University, Bethlehem, USA).

 Preface. Guillermo Cortiñas i On a differential intermediate value property. Matthias Aschenbrenner, Lou van den Dries, and Joris van der Hoeven Liouville closed $H$-fields are ordered differential fields where the ordering and derivation interact in a natural way and every linear differential equation of order $1$ has a nontrivial solution. (The introduction gives a precise definition.) For a Liouville closed $H$-field $K$ with small derivation we show that $K$ has the Intermediate Value Property for differential polynomials if and only if $K$ is elementarily equivalent to the ordered differential field of transseries. We also indicate how this applies to Hardy fields. 1–10 A remark on uniform expansion. Rafael Potrie For every $\mathcal{U} \subset \mathrm{Diff}^\infty_{\mathrm{vol}}(\mathbb{T}^2)$ there is a measure of finite support contained in $\mathcal{U}$ which is uniformly expanding. 11–21 The space of infinite partitions of $\mathbb{N}$ as a topological Ramsey space. Julián C. Cano and Carlos A. Di Prisco The Ramsey theory of the space of equivalence relations with infinite quotients defined on the set $\mathbb{N}$ of natural numbers is an interesting field of research. We view this space as a topological Ramsey space $(\mathcal{E}_\infty,\leq, r)$ and present a game theoretic characterization of the Ramsey property of subsets of $\mathcal{E}_{\infty}$. We define a notion of coideal and consider the Ramsey property of subsets of $\mathcal{E}_\infty$ localized on a coideal $\mathcal{H}\subseteq \mathcal{E}_{\infty}$. Conditions a coideal $\mathcal{H}$ should satisfy to make the structure $(\mathcal{E}_{\infty},\mathcal{H},\leq, r)$ a Ramsey space are presented. Forcing notions related to a coideal $\mathcal{H}$ and their main properties are analyzed. 23–48 Threshold Ramsey multiplicity for odd cycles. David Conlon, Jacob Fox, Benny Sudakov, and Fan Wei The Ramsey number $r(H)$ of a graph $H$ is the minimum $n$ such that any two-coloring of the edges of the complete graph $K_n$ contains a monochromatic copy of $H$. The threshold Ramsey multiplicity $m(H)$ is then the minimum number of monochromatic copies of $H$ taken over all two-edge-colorings of $K_{r(H)}$. The study of this concept was first proposed by Harary and Prins almost fifty years ago. In a companion paper, the authors have shown that there is a positive constant $c$ such that the threshold Ramsey multiplicity for a path or even cycle with $k$ vertices is at least $(ck)^k$, which is tight up to the value of $c$. Here, using different methods, we show that the same result also holds for odd cycles with $k$ vertices. 49–68 New bounds on Cantor maximal operators. Pablo Shmerkin and Ville Suomala We prove $L^p$ bounds for the maximal operators associated to an Ahlfors-regular variant of fractal percolation. Our bounds improve upon those obtained by I. Łaba and M. Pramanik and in some cases are sharp up to the endpoint. A consequence of our main result is that there exist Ahlfors-regular Salem Cantor sets of any dimension $> 1/2$ such that the associated maximal operator is bounded on $L^2(\mathbb{R})$. We follow the overall scheme of Łaba–Pramanik for the analytic part of the argument, while the probabilistic part is instead inspired by our earlier work on intersection properties of random measures. 69–86 Families of convex tilings. Richard Kenyon We study tilings of polygons $R$ with arbitrary convex polygonal tiles. Such tilings come in continuous families obtained by moving tile edges parallel to themselves (keeping edge directions fixed). We study how the tile shapes and areas change in these families. In particular we show that if $R$ is convex, the tile shapes can be arbitrarily prescribed (up to homothety). We also show that the tile areas and tile “orientations” determine the tiling. We associate to a tiling an underlying bipartite planar graph $\mathcal{G}$ and its corresponding Kasteleyn matrix $K$. If $\mathcal{G}$ has quadrilateral faces, we show that $K$ is the differential of the map from edge intercepts to tile areas, and extract some geometric and probabilistic consequences. 87–101 Higher Fano manifolds. Carolina Araujo, Roya Beheshti, Ana-Maria Castravet, Kelly Jabbusch, Svetlana Makarova, Enrica Mazzon, Libby Taylor, and Nivedita Viswanathan We address in this paper Fano manifolds with positive higher Chern characters, which are expected to enjoy stronger versions of several of the nice properties of Fano manifolds. For instance, they should be covered by higher dimensional rational varieties, and families of higher Fano manifolds over higher dimensional bases should admit meromorphic sections (modulo the Brauer obstruction). Aiming at finding new examples of higher Fano manifolds, we investigate positivity of higher Chern characters of rational homogeneous spaces. We determine which rational homogeneous spaces of Picard rank 1 have positive second Chern character, and show that the only rational homogeneous spaces of Picard rank 1 having positive second and third Chern characters are projective spaces and quadric hypersurfaces. We also classify Fano manifolds of large index having positive second and third Chern characters. We conclude by discussing conjectural characterizations of projective spaces and complete intersections in terms of these higher Fano conditions. 103–125 Wave maps into the sphere. Carlos E. Kenig In this note we discuss some geometric analogs of the classical harmonic functions on $\mathbb{R}^n$ and their associated evolutions. 127–135 Asymptotic mean value formulas for parabolic nonlinear equations. Pablo Blanc, Fernando Charro, Juan J. Manfredi and Julio D. Rossi In this paper we characterize viscosity solutions to nonlinear parabolic equations (including parabolic Monge–Ampère equations) by asymptotic mean value formulas. Our asymptotic mean value formulas can be interpreted from a probabilistic point of view in terms of dynamic programming principles for certain two-player, zero-sum games. 137–164 On fibrations and measures of irrationality of hyper-Khäler manifolds. Claire Voisin We prove some results on the fibers and images of rational maps from a hyper-Kähler manifold. We study in particular the minimal genus of fibers of a fibration into curves. The last section of this paper is devoted to the study of the rational map defined by a linear system on a hyper-Kähler fourfold satisfying numerical conditions similar to those considered by O'Grady in his study of fourfolds numerically equivalent to $\mathrm{K}3^{[2]}$. We extend his results to this more general context. 165–197 On conformally compact Einstein manifolds. Sun-Yung A. Chang and Yuxin Ge We survey some of the recent developments in the study of the compactness and uniqueness problems for some classes of conformally compact Einstein manifolds. 199–213