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Remarks on Symplectic Geometry

Jae-Hyun Yang

TL;DR

The article surveys five decades of progress in symplectic geometry, emphasizing the convexity of the moment map, the classification of symplectic and Hamiltonian actions, and the development of symplectic embedding problems and Gromov‑Witten theory. It surveys foundational results (Darboux, Weinstein, and Gromov), the Atiyah–Guillemin–Sternberg and Kirwan convexity theorems, and Delzant’s toric classification, then moves to modern themes such as symplectic embeddings, capacities, and the rich GW/FJRW framework and LG/CY correspondence. Key connections between pseudoholomorphic curves, Floer theory, and mirror symmetry are highlighted, along with the ongoing quest to classify and understand symplectic actions on manifolds. Overall, the paper situates how these interlinked themes shape current research in symplectic geometry, its applications to dynamics, topology, and mathematical physics, and the arising categorical structures like Fukaya categories and quantum cohomology.

Abstract

We survey the progress on the study of symplectic geometry past five decades. The survey focuses on the convexity properties of a moment map, the classification of symplectic actions, the symplectic embedding problems, and the theory of Gromov-Witten invariants.

Remarks on Symplectic Geometry

TL;DR

The article surveys five decades of progress in symplectic geometry, emphasizing the convexity of the moment map, the classification of symplectic and Hamiltonian actions, and the development of symplectic embedding problems and Gromov‑Witten theory. It surveys foundational results (Darboux, Weinstein, and Gromov), the Atiyah–Guillemin–Sternberg and Kirwan convexity theorems, and Delzant’s toric classification, then moves to modern themes such as symplectic embeddings, capacities, and the rich GW/FJRW framework and LG/CY correspondence. Key connections between pseudoholomorphic curves, Floer theory, and mirror symmetry are highlighted, along with the ongoing quest to classify and understand symplectic actions on manifolds. Overall, the paper situates how these interlinked themes shape current research in symplectic geometry, its applications to dynamics, topology, and mathematical physics, and the arising categorical structures like Fukaya categories and quantum cohomology.

Abstract

We survey the progress on the study of symplectic geometry past five decades. The survey focuses on the convexity properties of a moment map, the classification of symplectic actions, the symplectic embedding problems, and the theory of Gromov-Witten invariants.

Paper Structure

This paper contains 6 sections, 34 theorems, 170 equations.

Table of Contents

  1. Introduction
  2. The brief history of the terminology "Symplectic Geometry"
  3. Basic notions, definitions and examples
  4. Some fundamental theorems
  5. Convexity properties of the moment map
  6. Modern theory of symplectic geometry

Key Result

Theorem 2.1

(1) There exists exactly one geodesic joining two arbitrary points $\Omega_0,\,\Omega_1$ in ${\mathbb H}_g$. Let $R(\Omega_0,\Omega_1)$ be the cross-ratio defined by For brevity, we put $R_*=R(\Omega_0,\Omega_1).$ Then the symplectic length $\rho(\Omega_0,\Omega_1)$ of the geodesic joining $\Omega_0$ and $\Omega_1$ is given by where (2) For $M\in Sp(2g,\Bbb R)$, we set Then $R(\Omega_1,\Omega_

Theorems & Definitions (63)

  • Theorem 2.1
  • proof
  • Remark 2.2
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Theorem 3.4
  • Theorem 3.5
  • Definition 3.6
  • Definition 3.7
  • ...and 53 more