Second order rectifiability of varifolds of bounded mean curvature

We prove that the support of an m dimensional rectifiable varifold with a uniform lower bound on the density and bounded generalized mean curvature can be covered $$ {\mathscr {H}}^{m} $$ H m almost everywhere by a countable union of m dimensional submanifolds of class $$ {\mathcal {C}}^{2} $$ C 2 . The $$ {\mathcal {C}}^{2} $$ C 2 -regularity of the submanifolds is optimal.

On the closure of the Hodge locus of positive period dimension

Given $${{\mathbb {V}}}$$ V a polarizable variation of $${{\mathbb {Z}}}$$ Z -Hodge structures on a smooth connected complex quasi-projective variety S, the Hodge locus for $${{\mathbb {V}}}^\otimes $$ V ⊗ is the set of closed points s of S where the fiber $${{\mathbb {V}}}_s$$ V s has more Hodge tensors than the very general one. A classical result of Cattani, Deligne and Kaplan states that the Hodge locus for $${{\mathbb {V}}}^\otimes $$ V ⊗ is a countable union of closed irreducible algebraic subvarieties of S, called the special subvarieties of S for $${{\mathbb {V}}}$$ V . Under the assumption that the adjoint group of the generic Mumford–Tate group of $${{\mathbb {V}}}$$ V is simple we prove that the union of the special subvarieties for $${{\mathbb {V}}}$$ V whose image under the period map is not a point is either a closed algebraic subvariety of S or is Zariski-dense in S. This implies for instance the following typical intersection statement: given a Hodge-generic closed irreducible algebraic subvariety S of the moduli space $${{\mathcal {A}}}_g$$ A g of principally polarized Abelian varieties of dimension g, the union of the positive dimensional irreducible components of the intersection of S with the strict special subvarieties of $${{\mathcal {A}}}_g$$ A g is either a closed algebraic subvariety of S or is Zariski-dense in S.

A SEPARATION RESULT FOR COUNTABLE UNIONS OF BOREL RECTANGLES

We provide dichotomy results characterizing when two disjoint analytic binary relations can be separated by a countable union of ${\bf{\Sigma }}_1^0 \times {\bf{\Sigma }}_\xi ^0$ sets, or by a ${\bf{\Pi }}_1^0 \times {\bf{\Pi }}_\xi ^0$ set.

Zero-dimensional isomorphic dynamical models

By an assignment we mean a mapping from a Choquet simplex $K$ to probability measure-preserving systems obeying some natural restrictions. We prove that if $\unicode[STIX]{x1D6F7}$ is an aperiodic assignment on a Choquet simplex $K$ such that the set of extreme points $\mathsf{ex}K$ is a countable union $\bigcup _{n}E_{n}$, where each set $E_{n}$ is compact, zero-dimensional and the restriction of $\unicode[STIX]{x1D6F7}$ to the Bauer simplex $K_{n}$ spanned by $E_{n}$ can be ‘embedded’ in some topological dynamical system, then $\unicode[STIX]{x1D6F7}$ can be ‘realized’ in a zero-dimensional system.