Grant: Discovery Grants Program - Individual

Description for a wide audience (required)*This is an interdisciplinary proposal between pure mathematics, applied mathematics and computer science. Advanced methods from algebraic and differential geometry are used to create mathematical theory, algorithms and computer implementations for general systems of nonlinear partial differential equations. The algorithms identify and include missing constraints (integrability conditions) resulting from differentiating such systems. Such constraints restrict initial and boundary data, and are crucial in determination of analytical features and numerical solutions of such systems.*Incomplete systems, those that have missing constraints, arise frequently in geometric classification problems. This drew much attention from classical geometers such as Cartan and more recently by Olver and others. For example they arise in determination of transformations which left objects invariant (symmetries), or transformed one member of a class to another (equivalence transformations). Naturally a system is differentiated (prolonged) to obtain such integrability conditions. Developing criteria for how far a nonlinear system should be prolonged to include all such conditions has been notoriously difficult. Cartan conjectured, but was unable to prove, that his involutivity criteria resulted in such a finite prolongation. Kuranishi proved Cartan's conjecture, albeit under certain restrictions. Modern applications of geometric classification, such as the moving frames approach of Olver and collaborators, have led to complicated incomplete nonlinear systems. This has prompted the development of symbolic algorithms to include missing constraints, including work by the PI.*Incomplete systems commonly arise as higher index differential and partial differential algebraic equations (DAE and PDAE) where the index is the number of differentiations to include the missing constraints. Indeed the complexity of such DAE makes computers essential at every stage, from their formation, to completion, to numerical solution. This has prompted the development of powerful problem solving environments, such as MapleSim and SystemModeler. Indeed the PI's former student Wittkopf, is the main architect of the numerical engine of MapleSim. Such applications have motivated the current proposal, that of developing completion algorithms for approximate real systems, including inequalities. For example, we may need to specify that an unknown density is real and positive or that the position of a robot hand is constrained to move inside a cylinder. The most important part of this proposal is to build on some exciting breakthroughs in Semi-Definite Programming (SDP) and real numerical algebraic geometry to characterize real solutions of such nonlinear systems. Computer programs with user-friendly interfaces will make the results widely available.*