# Involutive Tableaux, Characteristic Varieties, and Rank-one Varieties in the Geometric Study of PDEs

## TL;DR

This paper explores advanced geometric concepts in PDEs, focusing on involutive tableaux, characteristic varieties, and rank-one varieties, providing a unified approach to key theorems in the field.

## Contribution

It introduces a new geometric interpretation of PDEs using Grassmann bundles and enhances Guillemin normal form to analyze involutivity of tableaux.

## Key findings

- Characterization of characteristic varieties and their incidence correspondence.
- Development of an enhanced Guillemin normal form for involutivity.
- Reinterpretation of PDE geometry through smooth sub-bundles of Grassmann bundles.

## Abstract

This expository monograph cuts a short path from the common, elementary background in geometry (linear algebra, vector bundles, and algebraic ideals) to the most advanced theorems about involutive exterior differential systems: (1) The incidence correspondence of the characteristic variety, (2) Guillemin normal form and Quillen's thesis, (3) The Integrability of Characteristics by Guillemin, Quillen, Sternberg, and Gabber, and (4) Yang's Hyperbolicity Criterion. To do so, the geometric theory of PDEs is reinterpreted as the study of smooth sub-bundles of the Grassmann bundle, whereby the rank-1 variety is emphasized. The primary computational tool is an enhanced formulation of Guillemin normal form that is equivalent to involutivity of tableaux.