Morse theory | Symplectic topology | 3-manifolds | Homology theory
In mathematics, Floer homology is a tool for studying symplectic geometry and low-dimensional topology. Floer homology is a novel invariant that arises as an infinite-dimensional analogue of finite-dimensional Morse homology. Andreas Floer introduced the first version of Floer homology, now called Lagrangian Floer homology, in his proof of the Arnold conjecture in symplectic geometry. Floer also developed a closely related theory for Lagrangian submanifolds of a symplectic manifold. A third construction, also due to Floer, associates homology groups to closed three-dimensional manifolds using the Yang–Mills functional. These constructions and their descendants play a fundamental role in current investigations into the topology of symplectic and contact manifolds as well as (smooth) three- and four-dimensional manifolds. Floer homology is typically defined by associating to the object of interest an infinite-dimensional manifold and a real valued function on it. In the symplectic version, this is the free loop space of a symplectic manifold with the symplectic action functional. For the (instanton) version for three-manifolds, it is the space of SU(2)-connections on a three-dimensional manifold with the Chern–Simons functional. Loosely speaking, Floer homology is the Morse homology of the function on the infinite-dimensional manifold. A Floer chain complex is formed from the abelian group spanned by the critical points of the function (or possibly certain collections of critical points). The differential of the chain complex is defined by counting the function's gradient flow lines connecting certain pairs of critical points (or collections thereof). Floer homology is the homology of this chain complex. The gradient flow line equation, in a situation where Floer's ideas can be successfully applied, is typically a geometrically meaningful and analytically tractable equation. For symplectic Floer homology, the gradient flow equation for a path in the loopspace is (a perturbed version of) the Cauchy–Riemann equation for a map of a cylinder (the total space of the path of loops) to the symplectic manifold of interest; solutions are known as pseudoholomorphic curves. The Gromov compactness theorem is then used to show that the differential is well-defined and squares to zero, so that the Floer homology is defined. For instanton Floer homology, the gradient flow equations is exactly the Yang–Mills equation on the three-manifold crossed with the real line. (Wikipedia).
Floer homology of Hamiltonians supported on subsets - Shira Tanny
Seminar in Analysis and Geometry Topic: Floer homology of Hamiltonians supported on subsets Speaker: Shira Tanny Affiliation: Member, School of Mathematics Date: December 14, 2021 Floer homology is a fundamental construction relating dynamical properties of Hamiltonian flows on symplecti
From playlist Mathematics
Edward Witten: "From Gauge Theory to Khovanov Homology Via Floer Theory”
Green Family Lecture Series 2017 "From Gauge Theory to Khovanov Homology Via Floer Theory” Edward Witten, Institute for Advanced Study Abstract: The goal of the lecture is to describe a gauge theory approach to Khovanov homology of knots, in particular, to motivate the relevant gauge the
From playlist Public Lectures
Family Floer theory and mirror symmetry - Mohammed Abouzaid
Workshop on Homological Mirror Symmetry: Methods and Structures Speaker Mohammed Abouzaid Title: Family Floer theory and mirror symmetry Affiliation: IAS Date: November 11, 2016 For more video, visit http://video.ias.edu
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From Gauge Theory to Khovanov Homology Via Floer Theory - Edward Witten
Workshop on Homological Mirror Symmetry: Emerging Developments and Applications Topic: From Gauge Theory to Khovanov Homology Via Floer Theory Speaker: Edward Witten Affiliation: IAS Date: March 15, 2017 For more video, visit http://video.ias.edu
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Geometry and topology of Hamiltonian Floer complexes in low-dimension - Dustin Connery-Grigg
Joint IAS/Princeton/Montreal/Paris/Tel-Aviv Symplectic Geometry Zoominar Topic: Geometry and topology of Hamiltonian Floer complexes in low-dimension Speaker: Dustin Connery-Grigg Affiliation: Université de Montreal Date: January 28, 2022 In this talk, I will present two results relating
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Floer Theory on Complex-Symplectic Manifolds - Semon Kirillovich Rezchikov
Short Talks by Postdoctoral Members Topic: Floer Theory on Complex-Symplectic Manifolds Speaker: Semon Kirillovich Rezchikov Affiliation: Member, School of Mathematics Date: September 22, 2022
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Floer Theory and Framed Cobordisms Between Exact Lagrangian Submanifolds - Noah Porcelli
Joint IAS/Princeton/Montreal/Paris/Tel-Aviv Symplectic Geometry Zoominar Topic: Floer Theory and Framed Cobordisms Between Exact Lagrangian Submanifolds Speaker: Noah Porcelli Affiliation: Imperial College London Date: February 24, 2023 Lagrangian Floer theory is a useful tool for study
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Lagrangian Floer theory in symplectic fibrations - Douglas Schultz
Princeton/IAS Symplectic Geometry Seminar Topic: Lagrangian Floer theory in symplectic fibrations Speaker: Douglas Schultz Affiliation: Rutgers University Date:April 27, 2017 For more info, please visit http://video.ias.edu
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Yi Xie - Surgery, Polygons and Instanton Floer homology
June 20, 2018 - This talk was part of the 2018 RTG mini-conference Low-dimensional topology and its interactions with symplectic geometry Many classical numerical invariants (including Casson invariant, Alexander polynomial and Jones polynomial) for 3-manifolds or links satisfy surgery fo
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Symplectic Instanton Homology of Knots and Links in 3-manifolds - David White
Joint IAS/Princeton/Montreal/Paris/Tel-Aviv Symplectic Geometry Zoominar Topic: Symplectic Instanton Homology of Knots and Links in 3-manifolds Speaker: David White Affiliation: North Carolina State University Date: February 10, 2023 Powerful homology invariants of knots in 3-manifolds
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Floer theory revisited - Mohammed Abouzaid
Princeton/IAS Symplectic Geometry Seminar Topic: Floer theory revisited Speaker: Mohammed Abouzaid Date: Thursday, February 4 Time/Room: 10:30am - 11:45am/S-101 I will describe a formalism for (Lagrangian) Floer theory wherein the output is not a deformation of the cohomology ring, but of
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Lagrangian Floer theory by Sushmita Venugopalan
J-Holomorphic Curves and Gromov-Witten Invariants DATE:25 December 2017 to 04 January 2018 VENUE:Madhava Lecture Hall, ICTS, Bangalore Holomorphic curves are a central object of study in complex algebraic geometry. Such curves are meaningful even when the target has an almost complex stru
From playlist J-Holomorphic Curves and Gromov-Witten Invariants
In search of Lagrangians with non-trivial Floer cohomology by Sushmita Venugopalan
DISCUSSION MEETING ANALYTIC AND ALGEBRAIC GEOMETRY DATE:19 March 2018 to 24 March 2018 VENUE:Madhava Lecture Hall, ICTS, Bangalore. Complex analytic geometry is a very broad area of mathematics straddling differential geometry, algebraic geometry and analysis. Much of the interactions be
From playlist Analytic and Algebraic Geometry-2018
A Heegaard Floer analog of algebraic torsion - Cagatay Kutluhan
Princeton/IAS Symplectic Geometry Seminar Topic: A Heegaard Floer analog of algebraic torsion Speaker: Cagatay Kutluhan Date: Thursday, April 21 The dichotomy between overtwisted and tight contact structures has been central to the classification of contact structures in dimension 3.
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Allison Moore - Essential Conway spheres and Floer homology via immersed curves
38th Annual Geometric Topology Workshop (Online), June 15-17, 2021 Allison Moore, Virginia Commonwealth University Title: Essential Conway spheres and Floer homology via immersed curves Abstract: We consider the problem of whether Dehn surgery along a knot in the three-sphere produces an
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Joanna NELSON - An Integral lift of contact homology
Cylindrical contact homology is arguably one of the more notorious Floer-theoretic constructions. The past decade has been less than kind to this theory, as the growing knowledge of gaps in its foundations has tarnished its claim to being a well-defined contact invariant. However, jointly
From playlist 2015 Summer School on Moduli Problems in Symplectic Geometry
A bordered approach to link Floer homology - Peter Ozsváth
IAS/PU-Montreal-Paris-Tel-Aviv Symplectic Geometry Topic: A bordered approach to link Floer homology Speaker: Peter Ozsváth Affiliation: Princeton University Date: July 10, 2020 For more video please visit http://video.ias.edu
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