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Research Report: some constructions in Polyhedral Symplectic Topology suggested by AI

Antonio Alfieri, Connor Novak

Abstract

We use large language models (LLM) to approach a question about Lagrangian smoothability proposed by Abouzaid et al. in "First Proof" arXiv:2602.05192.

Research Report: some constructions in Polyhedral Symplectic Topology suggested by AI

Abstract

We use large language models (LLM) to approach a question about Lagrangian smoothability proposed by Abouzaid et al. in "First Proof" arXiv:2602.05192.
Paper Structure (21 sections, 4 theorems, 43 equations)

This paper contains 21 sections, 4 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. Methodology
  3. Basic Set-Up
  4. Polyhedral Lagrangian surfaces
  5. Local primitive systems
  6. A basic example: products of polygonal curves
  7. Lagrangian and Hamiltonian smoothings
  8. Smoothing along the edges
  9. Normal form for a hinge and exact smoothing
  10. Local Obstructions at the vertices
  11. Cocycle obstruction (failed idea)
  12. Maslov index obstruction
  13. Topological Obstructions
  14. Local models at the vertices
  15. Normal form for the planes
  16. ...and 6 more sections

Key Result

Lemma 2.4

The cochain $c$ is a $1$--cocycle: $c\in Z^1(X_K^*;\mathbb{R})$.

Theorems & Definitions (14)

  • Definition 2.1: Lagrangian polyhedral surfaces
  • Definition 2.2: Local primitive system
  • Definition 2.3: Cocycle associated to a local primitive system
  • Lemma 2.4: Cocycle property
  • proof
  • Lemma 2.5
  • proof
  • Definition 2.6: Lagrangian smoothing
  • Definition 2.7: Hamiltonian Lagrangian smoothing
  • Lemma 3.1: Hinge normal form
  • ...and 4 more