Involution

W.M. Seiler is professor for computational mathematics (algorithmic algebra) at Kassel University. His research fields include differential equations, commutative algebra and mechanics. He is particularly interested in combining geometric and algebraic approaches. For many years, he has been an external developer for the computer algebra system MuPAD.

From the reviews: "The first book that provides a self-contained treatment of constructive and algorithmic ideas and methods in FTDE ... . the book Involution is unique, self-contained ... complete and, in addition, well-written with numerous instructive examples and comprehensive bibliography devoted to the constructive differential-geometric and computer-algebraic aspects of involutivity analysis of differential and polynomial equations and their applications. It can be recommended to researchers, postgraduate and advanced graduate students." (Vladimir P. Gerdt, Computeralgebra-Rundbrief, October, 2010) "This book deals with the formal theory of general ... systems of differential equations. ... starts with an introduction to a geometric approach to differential equations via the jet bundle formalism. ... this is a very interesting book with many instructive examples and strong emphasis placed on a constructive approach leading to effective algorithms. It is well written, so it is suitable for and recommended to anyone who is interested in computer-algebraic aspects of involutivity analysis of differential and polynomial equations and their applications." (Lech Slawik, Mathematical Reviews, Issue 2011 h) "The book targets the mathematically mature reader who can handle the frequent jumps between geometric and algebraic approaches. It is written at the postgraduate level ... . Seiler strikes a good balance between theory, algorithms, and applications. ... Seiler's book hopes to bridge two research communities: one of differential geometers concerned with the formal analysis of differential equations, the other of (computer) algebraists interested in computations with polynomial modules." (Willy Hereman, SIAM Review, Vol. 53 (3), 2011)

• Formal Geometry of Differential Equations.- Involution I: Algebraic Theory.- Completion to Involution.- Structure Analysis of Polynomial Modules.- Involution II: Homological Theory.- Involution III: Differential Theory.- The Size of the Formal Solution Space.- Existence and Uniqueness of Solutions.- Linear Differential Equations.- Miscellaneous.- Algebra.- Differential Geometry.