|Partial Differential Equations|
The formal theory of differential equations represents a combined algebraic/geometric approach to treat very general systems. "General" means in this context that the systems may be under- or overdetermined; no special assumptions are made about their structure. The notion of an involutive system plays a central role when dealing with such systems. Over the past years, I have been much concerned with completing arbitrary systems to involutive ones. This has lead to the design of some completion algorithms and their subsequent implementation in computer algebra systems. For linear systems we could combine geometric and algebraic approaches into a new hybrid algorithm that yields the full geometric information of the Cartan-Kuranishi theory with algebraic efficiency ; an extension to systems with polynomial nonlinearities is currently in progress. In recent times, the emphasis has shifted more towards understanding the properties of involutive systems and applications of the formal theory, for example in Mathematical Physics and Numerical Analysis.
There exists a number of approaches for dealing with general systems of differential equations. One for me very important aspect of the formal theory is its intrinsic geometric nature allowing for a coordinate free formulation. This leads to a very clear distinction between fundamental issues and mere technical problems in the algorithmic treatment. One of these fundamental issues is the Spencer cohomology of which I would like to obtain a deeper understanding. Some steps in this direction I did together with Larry Lambe (Rutgers/Bangor)  ; my current understanding is reflected in the review article  .
Currently, I am particularly interested in using geometric methods for studying differential equations. Together with my former Ph.D. student Dirk Fesser, I provided a rigorous formulation of the Vessiot theory, a dual, vector field based version of the Cartan-Kähler theory of exterior differential systems. In [58, 26] we showed that Vessiot's approach is valid, if and only if one deals with an involutive systems. Recently, I have started to apply the Vessiot theory for the analysis of geometric singularities; some of our ideas can be found in  . Currently, I concentrate together with Vladimir Gerdt (JINR, Dubna) mainly on ordinary differential equations (in particular, on the algorithmic detection of various kinds of geometric singularities), but the ultimate goal is to proceed to partial differential equations. As with most of my work, special emphasis lies on overdetermined and implicit systems. Together with Andreas Weber (Universität Bonn), I also study within the project Bifurcations and Singularities of Algebraic Differential Equations (BISADE) the relationship between bifurcations and singularities.
At the beginning, I studied various aspects of the problem of measuring the size of the (formal) solution space of a system of partial differential equations. For an involutive system this amounts essentially to determining its Cartan characters or equivalently its Hilbert polynomial (sometimes called differential dimension polynomial in this context). Some results in this direction are contained in [4, 6, 12, 48] . The first paper explains the basics and the other ones study several special situations in Lie symmetry theory. This includes the loss of generality in a symmetry reduction, a very general approach to non-classical symmetries and some (not very successful) thoughts on differential constraints.
Later, I started to study a lot the use of the formal theory in the analysis of mechanical systems with constraints. Some results are presented in the series of papers [8, 9, 17] . The first one deals with the classical Dirac theory and its relation to involution; the second one does the same for the Faddeev-Jackiw theory for affine Lagrangians (linear in the velocities). The third one discusses how can count degrees of freedom using formal methods. As a by-product it derives an equation for the Hilbert polynomials of differential equations connected by a differential relation. My goal is to provide on the basis of the formal theory a unified framework for the physical and the numerical theory of such systems (especially for field theories). In the paper  we showed for the special case of (possible explicitly time dependent) Hamiltonian systems with constraints how the constraint algorithm used in geometric mechanics can be united in an intrinsic manner with the completion algorithm of the formal theory. The extension to field theories is currently under investigation.
I have been generally interested in obtaining a deeper understanding of the meaning of involution. Formal integrability is a rather simple and straightforward concept: absence of integrability conditions. By why does one have sometimes to perform additional prolongations? The answer seems to lie in the algebraic properties of the symbol module which has lead me to enter more deeply into the realm of commutative algebra. The two articles [27, 28] provide some algebraic-combinatorial reasons. This combinatorial approach proves again and again to be surprisingly powerful. Another quite surprising discovery for me was that involution and not just formal integrability plays an important role in the numerical integration of general systems of differential equations. Some preliminary results and examples can be found in [18, 44, 50, 53] .
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