Spencer delta-cohomology, restrictions, characteristics and involutive symbolic PDEs

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

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-…

$\mathcal{D}$-几何希尔伯特概型——第一部分：对合性与稳定性

The $\mathcal{D}$-Geometric Hilbert Scheme -- Part I: Involutivity and Stability

本文通过引入$\mathcal{D}$-希尔伯特与$\mathcal{D}$-商函子构造PDE理想层的模空间，以斯宾塞稳定性为界性条件，并证明其与平坦联络的埃尔米特-杨-米尔斯度量的等价性，从而统一了几何PDE的稳定性理论。