Equivariant Lagrangian Floer Theory on Compact Toric Manifolds

This dissertation explores the field of Equivariant Lagrangian Floer Theory, specifically applied to compact symplectic toric manifolds with subtorus actions. The research introduces an equivariant version of Lagrangian Floer theory and demonstrates its application in proving that the set of Lagrangian torus fibers, along with weak bounding cochain data and non-vanishing equivariant Lagrangian Floer cohomology, forms a rigid analytic space. The work also shows how tropical geometry can be used to locate these Lagrangian torus fibers within the moment polytope. Furthermore, the study applies equivariant theory to establish that moment Lagrangian correspondences, induced by symplectic reduction, are unobstructed after bulk deformation, provided certain equivariant Kuranishi structures and compatible equivariant CF-perturbations exist.

Curriculum

The dissertation is structured to build foundational knowledge before delving into advanced topics. It begins with an overview of equivariant de Rham theory and compact symplectic toric manifolds, then progresses to the application of bulk deformation to Lagrangian correspondences, and finally introduces the equivariant Lagrangian Floer theory on compact symplectic toric manifolds.

1. Chapter 2
   • Equivariant de Rham theory
   • Preliminary definitions
   • Equivariant cohomology
   • Equivariant integration along the fiber
   • Equivariant Thom forms
2. Chapter 3
   • Compact symplectic toric manifolds
   • Hamiltonian group actions
   • Constructions of toric manifolds from Delzant polytopes
3. Chapter 4
   • Ordinary Lagrangian Floer theory on compact toric manifolds
   • The A∞algebra associated to a Lagrangian submanifold
   • Bulk deformations
   • Moment Lagrangian correspondences are unobstructed after bulk deformation
4. Chapter 5
   • The equivariant A∞algebra associated with a Lagrangian torus fiber
   • The setup
   • The moduli space of pseudoholomorphic discs
   • The equivariant A∞algebra associated to a Lagrangian submanifold
   • Spectral sequences
5. Chapter 6
   • CritG(PD) is a rigid analytic space
   • Examples
6. Chapter 7
   • Equivariant Hamiltonian isotopy invariance
   • The cochain complex CFG(L,b,H),δH G
   • The Floer continuation map f
   • The map g
   • Proof of Hamiltonian isotopy invariance
7. Chapter 8
   • Equivariant Kuranishi structures
   • Equivariant Kuranishi structures
   • Equivariant CF-perturbations
   • Equivariant integration along the fiber
   • Equivariant Stokes’ theorem and the smooth correspondences

Careers

While this dissertation focuses on theoretical mathematics, the skills developed in advanced abstract reasoning, complex problem-solving, and rigorous proof techniques are highly transferable.

• Researcher
• Academic
• Data Scientist
• Analyst