Thursday May 2nd, 2024 | |
| Time: | 1:30 PM |
| Title: | Revisiting Localization, Periodicity and Galois Symmetry |
| Speaker: | Runjie Hu, Stony Brook University |
| Location: | Math 5-127 |
| Abstract: | |
| It is known that two complex algebraic varieties can be algebraically isomorphic but notrnbe homeomorphic. Such examples can be obtained by changing the coefficients of therndefining equations by some field automorphism of a finite extension of the rationals Q.rnThis dissertation aims to understand how the entire Galois group of Q-bar, the algebraicrnclosure of Q, changes the underlying manifold structures of smooth complex varietiesrndefined by equations with coefficients in Q-bar. It is known by the theory of finite coveringrnspaces (étale theory) that the Galois action does not change that aspect of the homotopyrntype determined by finite group theory (the profinite homotopy type). Thus we can use thernknown theory of manifolds in a given homotopy type to study the Galois conjugates ofrnalgebraic varieties in a given étale homotopy type. We study three aspects of this problem:rn(1) what algebraic-topological data is sufficient to specify a topological manifold in arnhomotopy type; (2) what might be the étale construction for manifolds; (3) how might onernexpress the Galois action in terms of the algebraic-topological data. We suggest an approachrnusing the study in (2) in order to propose a geometric interpretation of the question in (3). | |