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If $\bar\rho$ is an automorphic modulo $p$ Galois representation, it is natural to wonder if automorphic points are Zariski dense in the deformation space of $\bar\rho$. We prove new results in this direction in the case of a unitary group…

We study a notion of indecomposability in differential algebraic groups which is inspired by both model theory and differential algebra. After establishing some basic definitions and results, we prove an indecomposability theorem for…

We study the discrete dynamics of standard (or left) polynomials $f(x)$ over division rings $D$. We define their fixed points to be the points $\lambda \in D$ for which $f^{\circ n}(\lambda)=\lambda$ for any $n \in \mathbb{N}$, where…

The Furstenberg-Zimmer structure theorem for $\mathbb{Z}^d$ actions says that every measure-preserving system can be decomposed into a tower of primitive extensions. Furstenberg and Katznelson used this analysis to prove the…

We study the \emph{picture space} $X^d(G)$ of all embeddings of a finite graph $G$ as point-and-line arrangements in an arbitrary-dimensional projective space, continuing previous work on the planar case. The picture space admits a natural…

Given a polarizable $\mathbb{Z}$-variation of Hodge structures $\mathbb{V}$ over a complex smooth quasi-projective base $S$, a classical result of Cattani, Deligne and Kaplan says that its Hodge locus (i.e. the locus where exceptional Hodge…

Given an algebraic family $f\colon\Lambda\times \mathbb{A}^2 \to \Lambda\times \mathbb{A}^2$ of plane polynomial automorphisms of H\'enon type parameterized by a quasi-projective curve, defined over a number field $\mathbb{K},$ we…

We obtain a polynomial upper bound in the finite-field version of the multidimensional polynomial Szemer\'{e}di theorem for distinct-degree polynomials. That is, if $P_1, ..., P_t$ are nonconstant integer polynomials of distinct degrees and…

We use recent developments in local entropy theory to prove that chaos in dynamical systems implies the existence of complicated structure in the underlying space. Earlier Mouron proved that if $X$ is an arc-like continuum which admits a…

We prove a $p$-adic analogue of the Andr\'{e}-Oort conjecture for subvarieties of the universal abelian varieties containing a dense set of special points. Let $g$ and $n$ be integers with $n \geq 3$ and $p$ a prime number not dividing $n$.…

A complex hyperplane arrangement $\mathcal{A}$ is said to be decomposable if there are no elements in the degree 3 part of its holonomy Lie algebra besides those coming from the rank 2 flats. When this purely combinatorial condition is…

Let $f:\mathbb{A}^N\to\mathbb{A}^N$ be a regular endomorphism of algebraic degree $d\geq2$ (i.e., $f$ extends to an endomorphism on $\mathbb{P}^N$ of algebraic degree $d$) defined over a number field. We prove that if the set of cyclotomic…

Let $K$ be a function field of characteristic $p\geq0$ or a number field over which the $abc$ conjecture holds, and let $\phi(x)=x^d+c \in K[x]$ be a unicritical polynomial of degree $d\geq2$ with $d \not\equiv 0,1\pmod{p}$. We completely…

We show that two automorphisms of an affine surface with dynamical degree strictly larger than 1 share a Zariski dense set of periodic points if and only if they have the same periodic points. We construct canonical heights for these…

It is proved that if a finite $p$-soluble group $G$ admits an automorphism $\varphi$ of order $p^n$ having at most $m$ fixed points on every $\varphi$-invariant elementary abelian $p'$-section of $G$, then the $p$-length of $G$ is bounded…

We prove that, over any field, the dimension of the indeterminacy locus of a rational transformation $f$ of $P^n$ which is defined by monomials of the same degree $d$ with no common factors is at least $(n-2)/2$, provided that the degree of…

Let X be an irreducible n-dimensional projective variety in CP^N with arbitrary singular locus. We prove that the L2-(p,1)-d-bar cohomology groups (with respect to the Fubini-Study metric) of the regular part of X are finite dimensional.

Any discrete distribution with support on $\{0,\ldots, d\}$ can be constructed as the distribution of sums of Bernoulli variables. We prove that the class of $d$-dimensional Bernoulli variables $\boldsymbol{X}=(X_1,\ldots, X_d)$ whose sums…

In this follow-up paper, we again inspect a surprising relationship between the set of fixed points of a polynomial map $\varphi_{d, c}$ defined by $\varphi_{d, c}(z) = z^d + c$ for all $c, z \in \mathcal{O}_{K}$ or $\in \mathbb{Z}_{p}$ or…

Let $X$ be an affine or a projective variety defined over a number field $K$ and $\varphi:{\bf C}\to X({\bf C})$ be a holomorphic map with Zariski dense image. We estimate the number of rational points of height bounded by $H$ in the image…