Programs

Revisiting Localization, Periodicity and Galois Symmetry

This dissertation explores the relationship between algebraic geometry and topology, specifically how the Galois group of the algebraic closure of rational numbers (Q) affects the underlying manifold structures of complex algebraic varieties defined over Q. While algebraic isomorphisms can exist between varieties, their topological structures (like homeomorphism type) might differ. The research uses concepts like 'profinite completion' to understand these changes, as the Galois action preserves this 'profinite homotopy type'. The study investigates what algebraic-topological data is sufficient to define a manifold within a given homotopy type, how to construct these manifolds algebraically ('étale construction'), and how the Galois action can be expressed using this data.