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Particularly, is there any connection between formal/first-order/infinitesimal deformation theory and perturbation theory? Both subjects involve "perturbing" some structure at a point, so intuitively it is quite similar.

Forgive me if this question is naive, as I don't know much about either area.

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    $\begingroup$ Kodaira Spencer theory generalizes from the deformation theory of complex structures to the deformation theory of a wide variety of geometric structures on manifolds, often defined by partial differential equations, particularly the theory of Lie pseudogroups. Much of the machinery has some interpretation for more general systems of partial differential equations, particular Spencer cohomology; see Bryant et. al., Exterior Differential Systems. $\endgroup$
    – Ben McKay
    Commented Mar 10, 2022 at 16:17
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    $\begingroup$ Kumpera, A.; Spencer, D. C. (1972), Lie Equations: Volume I, General Theory, AM-73, Annals of Mathematical Studies, Princeton University Press, ISBN 978-0-6910-8111-3; pbk Kumpera, A.; Spencer, D. C. (1974), Systems of Linear Partial Differential Equations and Deformation of Pseudogroup Structures, Les Presses de l'Université de Montréal $\endgroup$
    – Ben McKay
    Commented Mar 10, 2022 at 16:29

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