Local Lagrangian and Fixed-Point Floer (Co)homologies

This dissertation presents two related research projects in mathematics. The first project develops a general framework for calculating specific types of invariants in Lagrangian Floer theory using local data. This work extends previous research by introducing a spectral sequence that helps compute these invariants, with applications demonstrated in the study of affine varieties. The second project focuses on a different kind of invariant related to symplectomorphisms, specifically examining families of singularities in Milnor fibrations. It offers a new proof for a conjecture by Zariski concerning the constancy of certain properties in these singularity families, and also recovers a result by Varchenko. While these two projects explore different areas, they are connected by the inspiration for the first project's definitions, which stem from singularities of algebraic varieties. Future work aims to use the developed spectral sequence for further computations. The dissertation includes foundational concepts relevant to both projects before delving into the specifics of each part, which can be read independently.

Curriculum

The dissertation is structured into two main parts, preceded by foundational concepts. Part one focuses on local Lagrangian Floer theory and spectral sequences, while part two deals with fixed-point Floer cohomology and isolated hypersurface singularities.

1. Foundational Concepts
   • Symplectic and Contact Geometry
   • Minimal Degeneracy Definitions
   • Lagrangian Quasi-Minimal Degeneracy
   • Clean Intersections
   • Isolated Hypersurface Singularities and Milnor Fibrations
   • Liouville Manifolds
   • Algebraic Invariants and Open Book Decompositions
   • Fixed-Point Floer Cohomology
2. Part 1: Local Lagrangian and Fixed-Point Floer Cohomologies
   • A Spectral Sequence
   • Applications to Affine Varieties
3. Part 2: Fixed-Point Floer Cohomology and Singularities
   • Adjacent Singularities and Cobordisms
   • Novel Proof of Zariski's Conjecture
   • Constant Families of Singularities
4. Appendices
   • Gradings and Conley-Zehnder Index
   • Fixed-Point Floer Cohomology and Symplectic Cohomology
   • Systems of Adjacent Singularities

Careers

The dissertation focuses on advanced theoretical mathematics research. Potential career paths for individuals with this level of expertise typically involve academic research and teaching positions in universities, or specialized research roles in scientific institutions and industries that require advanced mathematical modeling and problem-solving skills.

• University Professor
• Researcher (Academia/Industry)
• Applied Mathematician