Floer Homologies

• Grant number: 25/12539-9
• Support Opportunities: Scholarships in Brazil - Master
• Start date: August 01, 2025
• End date: July 31, 2027
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Renato Ferreira de Velloso Vianna
• Grantee: Pedro Coelho Salgado Dias
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil
• Associated research grant: 24/01351-6 - Lagrangian submanifolds: open Gromov-Witten theory and Mirror Symmetry, AP.JP

Abstract

The 1980s were a revolutionary decade in simplectic geometry with the introduction of the use of J-holomorphic curves by Gromov, the development of Gromov-Witten theory and the theory developed by Floer.Floer's homologies were inspired by Morse-Witten-Smale theory and the development of J-holomorphic curves. They were used to address Arnold's conjecture, resp. Arnold-Givental conjecture, which provides a lower bound for the number of fixed points of a non-degenerate Hamiltonian simplectomorphism, resp. the intersection of a Lagrangian subvariety with a transverse Hamiltonian shift of it.In addition to Hamiltonian and Lagrangian Floer homologies, several versions of Floer homologies have been developed over time, such as: Heegard Floer homology, Rabinowitz Floer homology, sutured Floer homology, simplicial homology, immersed contact homology, wrapped Floer homology, addressing questions in dynamics, simplectic topology and knot theory.In this project, we will study Hamiltonian and Lagrangian Floer homologies in detail, covering the technical details involved in their definitions. This includes the interpretation of the Floer complex as generated by critical points of an action functional and of the differential as the gradient trajectories of the same functional, as well as, Gromov compactness, transversality, Conley-Zhender and Maslov indices.Time permitting, we will explore ideas related to these homologies and/or definitions of other Floer homologies.