Let $F$ be a non-Archimedean locally compact field, and let $G$ be a split connected reductive group over $F$. For a parabolic subgroup $Q\subset G$ and a ring $L$, we consider the $G$-representation on the $L$-module $$C^{\infty }(G/Q,L)/{\sum }_{Q\text{'}⊋Q}{C}^{\infty }(G/Q\text{'},L).(\ast )$$ Let $I\subset G$ denote an Iwahori subgroup. We define a certain free finite rank-$L$ module $\mathfrak{M}$ (depending on $Q$; if $Q$ is a Borel subgroup, then (∗) is the Steinberg representation and $\mathfrak{M}$ is of rank $1$) and construct an $I$-equivariant embedding of (∗) into $C^{\infty }(I,\mathfrak{M})$. This allows the computation of the $I$-invariants in (∗). We then prove that if $L$ is a field with characteristic equal to the residue characteristic of $F$ and if $G$ is a classical group, then the $G$-representation (∗) is irreducible. This is the analogue of a theorem of Casselman (which says the same for $L=\mathbb{C}$); it had been conjectured by Vignéras.

Herzig (for $G={GL}_n\left(F\right)$) and Abe (for general $G$) have given classification theorems for irreducible admissible modulo $p$ representations of $G$ in terms of supersingular representations. Some of their arguments rely on the present work.

We provide examples of families of (log) smooth canonically polarized varieties, including smooth weighted pointed curves and smooth hypersurfaces in ${\mathbb{P}}^3$ with large degree such that the Chow semistable limits under distinct pluricanonical embeddings do not stabilize.

A classical and widely used lemma of Erdős and Szekeres asserts that for every $n$ there exists $N$ such that every $N$-term sequence $\underline{a}$ of real numbers contains an $n$-term increasing subsequence or an $n$-term nonincreasing subsequence; quantitatively, the smallest $N$ with this property equals $(n-1{)}^2+1$. In the setting of the present paper, we express this lemma by saying that the set of predicates $\mathit{\Phi }=\{x_1<x_2,x_1\ge x_2\}$ is Erdős–Szekeres with Ramsey function ${ES}_{\mathit{\Phi }}\left(n\right)=(n-1{)}^2+1$.

In general, we consider an arbitrary finite set $\mathit{\Phi }=\{{\Phi }_1,\dots ,{\Phi }_m\}$ of semialgebraic predicates, meaning that each ${\Phi }_j={\Phi }_j(x_1,\dots ,x_k)$ is a Boolean combination of polynomial equations and inequalities in some number $k$ of real variables. We define $\mathit{\Phi }$ to be Erdős–Szekeres if for every $n$ there exists $N$ such that each $N$-term sequence $\underline{a}$ of real numbers has an $n$-term subsequence $\underline{b}$ such that at least one of the ${\Phi }_j$ holds everywhere on $\underline{b}$, which means that ${\Phi }_j(b_{i_1},\dots ,b_{i_k})$ holds for every choice of indices $i_1,i_2,\dots ,i_k$, $1\le i_1<i_2<\cdots <i_k\le n$. We write ${ES}_{\mathit{\Phi }}\left(n\right)$ for the smallest $N$ with the above property.

We prove two main results. First, the Ramsey functions in this setting are at most doubly exponential (and sometimes they are indeed doubly exponential): for every $\mathit{\Phi }$ that is Erdős–Szekeres, there is a constant $C$ such that ${ES}_{\mathit{\Phi }}\left(n\right)\le 2^{2^{Cn}}$. Second, there is an algorithm that, given $\mathit{\Phi }$, decides whether it is Erdős–Szekeres; thus, $1$-dimensional Erdős–Szekeres-style theorems can in principle be proved automatically.

We regard these results as a starting point in investigating analogous questions for $d$-dimensional predicates, where instead of sequences of real numbers, we consider sequences of points in ${\mathbb{R}}^d$ (and semialgebraic predicates in their coordinates). This setting includes many results and problems in geometric Ramsey theory, and it appears considerably more involved. Here we prove a decidability result for algebraic predicates in ${\mathbb{R}}^d$ (i.e., conjunctions of polynomial equations), as well as for a multipartite version of the problem with arbitrary semialgebraic predicates in ${\mathbb{R}}^d$.

This paper describes the intrinsic geometry of a leaf $\mathcal{A}\left(L\right)$ of the absolute period foliation of the Hodge bundle $\Omega {\overline{\mathcal{M}}}_g$: its singular Euclidean structure, its natural foliations, and its discretized Teichmüller dynamics. We establish metric completeness of $\mathcal{A}\left(L\right)$ for general $g$ and then turn to a study of the case $g=2$. In this case the Euclidean structure comes from a canonical meromorphic quadratic differential on $\mathcal{A}\left(L\right)\cong \mathbb{H}$ whose zeros, poles, and exotic trajectories are analyzed in detail.