Relativization of sensitivity in minimal systems

Let $\unicode[STIX]{x1D70B}:(X,T)\rightarrow (Y,S)$ be an extension between minimal systems; we consider its relative sensitivity. We obtain that $\unicode[STIX]{x1D70B}$ is relatively $n$-sensitive if and only if the relative $n$-regionally proximal relation contains a point whose coordinates are distinct; and the structure of $\unicode[STIX]{x1D70B}$ which is relatively $n$-sensitive but not relatively $(n+1)$-sensitive is determined. Let ${\mathcal{F}}_{t}$ be the families consisting of thick sets. We introduce notions of relative block ${\mathcal{F}}_{t}$-sensitivity and relatively strong ${\mathcal{F}}_{t}$-sensitivity. Let $\unicode[STIX]{x1D70B}:(X,T)\rightarrow (Y,S)$ be an extension between minimal systems. Then the following Auslander–Yorke type dichotomy theorems are obtained: (1) $\unicode[STIX]{x1D70B}$ is either relatively block ${\mathcal{F}}_{t}$-sensitive or $\unicode[STIX]{x1D719}:(X,T)\rightarrow (X_{\text{eq}}^{\unicode[STIX]{x1D70B}},T_{\text{eq}})$ is a proximal extension where $(X_{\text{eq}}^{\unicode[STIX]{x1D70B}},T_{\text{eq}})\rightarrow (Y,S)$ is the maximal equicontinuous factor of $\unicode[STIX]{x1D70B}$. (2) $\unicode[STIX]{x1D70B}$ is either relatively strongly ${\mathcal{F}}_{t}$-sensitive or $\unicode[STIX]{x1D719}:(X,T)\rightarrow (X_{d}^{\unicode[STIX]{x1D70B}},T_{d})$ is a proximal extension where $(X_{d}^{\unicode[STIX]{x1D70B}},T_{d})\rightarrow (Y,S)$ is the maximal distal factor of $\unicode[STIX]{x1D70B}$.

On function field Mordell–Lang and Manin–Mumford

We give a reduction of the function field Mordell–Lang conjecture to the function field Manin–Mumford conjecture, for abelian varieties, in all characteristics, via model theory, but avoiding recourse to the dichotomy theorems for (generalized) Zariski geometries. Additional ingredients include the “Theorem of the Kernel”, and a result of Wagner on commutative groups of finite Morley rank without proper infinite definable subgroups. In positive characteristic, where the main interest lies, there is one more crucial ingredient: “quantifier-elimination” for the corresponding [Formula: see text] where [Formula: see text] is a saturated separably closed field.

The graph-theoretic approach to descriptive set theory

We sketch the ideas behind the use of chromatic numbers in establishing descriptive set-theoretic dichotomy theorems.