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.

Curriculum

This dissertation delves into advanced mathematical concepts related to algebraic topology and number theory. It is structured into several chapters covering preliminary theories, specific mathematical frameworks, and applications to algebraic varieties.

1. Chapter 2: Preliminaries
   • Surgery Theory (Manifolds in a Homotopy Type, L Groups, L Spectra)
   • Profinite Completion and Galois Symmetry (Localization and Completion, Sites and Hypercoverings, Adams Conjecture and Sullivan’s Proof)
   • A Priori Invariants by Periods (Z*/n*-Manifolds, Cohomological and K-Theoretical A Priori Invariants, Applications to Surgery Theory)
2. Chapter 3: L-theory with Z*/n* coefficient
   • Z*/n* Chains and their bordisms
   • Product of Z*/n* Chains
   • Product Structure of L-Theory with Z*/n* coefficient (Symmetric and Quadratic Case, Normal Case)
3. Chapter 4: L-theory Characteristic Classes and Bundle Lifting Problem
   • L-theory Characteristic Classes and Splittings of L-spectra at Prime 2 (Quadratic L-spectrum, Symmetric L-spectrum, Normal L-spectrum)
   • Relations of Characteristic Classes
   • Coproducts of Characteristic Classes
   • Characteristic Classes of Bundle Theory
   • L-theory at Odd Primes
4. Chapter 5: Branched Covering and Profinite Completion
   • Pseudomanifold and Branched Covering (Simplicial Pseudomanifold, Branched Covering)
   • Riemann Existence Theorem for Pseudomanifold
5. Chapter 6: Galois Symmetry on Simply Connected Topological Manifolds
   • Simply Connected Profinite Manifold Structure (Simply Connected p-adic Poincaré Space, Manifold Structure at Odd Primes, Manifold Structure at Prime 2, Etale Manifold Structure Set)
   • Galois Symmetry on Varieties
   • Conjectures and Questions

Careers

Graduates with a PhD in Mathematics from Stony Brook University are prepared for advanced research and academic positions. The skills developed, particularly in abstract algebra, topology, and theoretical mathematics, are applicable in various research-intensive fields.

• University Professor
• Research Scientist
• Mathematician
• Data Scientist
• Cryptographer