Compatible maps and Borel maps of involutive structures

In the present project, we study symmetries and Borel mappings of so-called involutive systems of linear partial differential equations one can create such a system from any linear partial differential equation that has the same solutions. The two objectives are complementary in some ways, but both serve to better understand the solutions of partial differential equation systems. A symmetry of a system of linear differential equations is, roughly speaking, a transformation of the underlying coordinates that converts solutions of the system into solutions. Symmetries can be studied using both analytical and geometric methods; in our project, we focus on the second approach. For special involutive systems (the so-called Cauchy-Riemann or CR systems), the theory is already very well developed, but the methods developed for their study use a connection between solutions of these systems and mappings that are generally not available to us. We are interested in developing methods that can also be applied to other involutive systems. We want to show that similar to CR systems, symmetries for most involutive systems only possess finitely many degrees of freedom, and we want to characterize the conditions that lead to the existence of an infinite-dimensional family of such symmetries. Borel mappings represent a relationship between solutions of involutive systems and so-called formal solutions. The name is chosen because of a theorem of Borel, which states that every formal power series in a variable is the Taylor series of a smooth function. Similarly, one can ask whether the power series developments of solutions are subject to other, actually interesting constraints besides the obvious constraints (this leads to so-called formal solutions). For example, for the usual CR differential equations in the plane, all formal solutions that are Taylor developments of actual solutions are convergent. For general involutive systems, there are sufficient conditions that show the validity of Borel`s theorem or that make the analogy to the CR differential equations in the plane. We are interested in whether there are geometric possibilities to formulate necessary conditions for such results for systems which possess enough regularity.