Algebraic Varieties and Diffieties

In modern mathematics a sector called the Theory of Algebraic Equations does not exist. Yet algebraic equations are studied by Algebraic Geometry, which constitutes the modern form of the theory of algebraic equations. This is the outcome of a fairly long evolutionary process showing that the right object to study are not the sets of concrete equations but the ideals generated by them and the corresponding algebraic varieties. In other words the concept of an algebraic variety—more exactly the spectrum of a commutative algebra—is the conceptually right answer to the question of what a system of algebraic equations is. Thinking about Groetendieck's schemes it is easy to understand that the answer to this and many other similar questions are not so obvious and naive as one may expect at first sight.
Taking this into account, it is natural to expect that the conceptually right answer to the question of what a system of non-linear PDEs is will be even less obvious than the previous question. And so it is. In particular this is due to the fact that when building the ideal corresponding to a set of differential relations one must take into account not only the algebraic consequences of such relations but also their differential consequences. It is of fundamental importance the possibility of working with all the consequences—both algebraic and differential—of the original relations, also from the viewpoint of the logic completeness of the theoretical scheme. Following this path, it is possible to discover the concept of a diffiety, a new geometrical object playing the same role in the modern theory of PDEs as the algebraic varieties play for algebraic equations. By their very nature diffieties are carrier of many geometrical structures, the most important of which is the infinite-order contact structure—the so-called Cartan distribution—containing all the information about the system of original PDEs. Form the viewpoint of the geometrical theory of non-linear PDEs the study of a concrete system of such equations is nothing but the study of the geometrical structures on the corresponding diffiety. This way, the theory of non-linear PDEs takes the rigorous and concrete form of the study of general properties of diffieties and natural operations on them.