

Published

Vol. 64, no. 1 (2022)

Front matter 
Preface.  i 
On a differential intermediate value property.
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.
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.
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.
The Ramsey number $r(H)$ of a graph $H$ is the minimum $n$ such that any
twocoloring 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 twoedgecolorings 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.
We prove $L^p$ bounds for the maximal operators associated to an
Ahlforsregular 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 Ahlforsregular
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.
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.
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.
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.
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 twoplayer, zerosum games.

137–164 
On fibrations and
measures of irrationality of hyperKhäler manifolds.
We prove some results on the fibers and images of rational maps from a
hyperKähler 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
hyperKähler 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.
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 
Published by the Unión Matemática Argentina
ISSN 16699637 (online), ISSN 00416932 (print)
Registro Nacional de la Propiedad Intelectual nro. 180.863
Contact: revuma(at)criba.edu.ar