Tag - Floer theory

An exceptionally gifted mathematician and an extremely complex person, Floer exhibited, as one friend put it, a 'radical individuality'. He viewed the world around him with a singularly critical way of thinking and a quintessential disregard for convention. Indeed, his revolutionary mathematical ideas, contradicting conventional wisdom, could only be inspired by such impetus, and can only be understood in this context.

Poincaré's research on the Three Body Problem laid the foundations for the fields of dynamical systems and symplectic geometry. From whence the ancestral trail follows Marston Morse and Morse theory, Vladimir Arnold and the Arnold conjectures, through to breakthroughs by Yasha Eliashberg. Likewise, Charles Conley and Eduard Zehnder on the Arnold conjectures, Mikhail Gromov's theory of pseudoholomorphic curves, providing a new and powerful tool to study symplectic geometry, and Edward Witten's fresh perspective on Morse theory. And finally, Andreas Floer, who counter-intuitively combined all of this, hitting the "jackpot" with what is now called Floer theory.

Given a family of Lagrangian tori with full quantum corrections, the non-archimedean SYZ mirror construction uses the family Floer theory to construct a non-archimedean analytic space with a global superpotential. In this talk, we will first briefly review the construction. Then, we will apply it to the Gross’s fibrations. As an application, we can compute all the non-trivial open GW invariants for a Chekanov-type torus in ℂPⁿ or ℂP^r×ℂP^n-r. When n=2, r=1, we retrieve the previous results of Auroux an Chekanov-Schlenk without finding the disks explicitly. It is also compatible with the Pascaleff-Tonkonog’s work on Lagrangian mutations.

The purpose of this talk is to explore how Lagrangian Floer homology groups change under (non-Hamiltonian) symplectic isotopies on a (negatively) monotone symplectic manifold (M,ω) satisfying a strong non-degeneracy condition. More precisely, given two Lagrangian branes L,L′, consider family of Floer homology groups HF(ϕ_v(L),L′), where v ∈ H¹(M,ℝ) and ϕ_v is the time-1 map of a symplectic isotopy with flux v. We show how to fit this collection into an algebraic sheaf over the algebraic torus H¹(M,𝔾_m). The main tool is the construction of an "algebraic action" of H¹(M,𝔾_m) on the Fukaya category. As an application, we deduce the change in Floer homology groups satisfy various tameness properties, for instance, the dimension is constant outside an algebraic subset of H¹(M,𝔾_m). Similarly, given closed 1-form α, which generates a symplectic isotopy denoted by ϕ^t_α, the Floer homology groups HF(ϕ^t_α(L),L′) have rank that is constant in t, with finitely many possible exceptions.