Involutive tableaux, characteristic varieties, and rank-one varieties in the geometric study of PDEs

Tom 117 / 2019

Abraham D. Smith Banach Center Publications 117 (2019), 57-112 MSC: Primary: 58A15; Secondary 35A27, 35A30. DOI: 10.4064/bc117-3


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.


  • Abraham D. SmithDepartment of Mathematics, Statistics and Computer Science
    University of Wisconsin-Stout
    Menomonie, Wisconsin 54751-2506, USA

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