A moduli-theoretic framework for PDEs via D-Hilbert schemes and Spencer stability is introduced, but the advertised refinement of Donaldson-Uhlenbeck-Yau is a restatement of the classical result.
Homological Methods in Equations of Mathematical Physics
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abstract
These lecture notes are a systematic and self-contained exposition of the cohomological theories naturally related to partial differential equations: the Vinogradov C-spectral sequence and the C-cohomology, including the formulation in terms of the horizontal (characteristic) cohomology. Applications to computing invariants of differential equations are discussed. The lectures contain necessary introductory material on the geometric theory of differential equations and homological algebra.
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The $\mathcal{D}$-Geometric Hilbert Scheme -- Part I: Involutivity and Stability
A moduli-theoretic framework for PDEs via D-Hilbert schemes and Spencer stability is introduced, but the advertised refinement of Donaldson-Uhlenbeck-Yau is a restatement of the classical result.