Symbolic-numeric algorithms and applications for systems of differential polynomial equations and inequalities

面向广大读者的描述（必填项）***这是一个纯数学、应用数学和计算机科学之间的跨学科的建议。先进的方法从代数和微分几何是用来创建数学理论，算法和计算机实现的一般系统的非线性偏微分方程。该算法识别并包括由微分系统导致的缺失约束（可积性条件）。这些约束限制了初始数据和边界数据，对于确定此类系统的解析特征和数值解至关重要。不完全系统，即缺少约束的系统，在几何分类问题中经常出现。这引起了古典几何学家的极大关注，比如卡坦，以及最近的奥利弗等人。例如，它们出现在使对象保持不变（对称）的转换的确定中，或者将类的一个成员转换为另一个成员（等价转换）。自然，一个系统被微分（延长）以得到这样的可积条件。制定一个非线性系统应扩展到何种程度以包括所有这些条件的标准是出了名的困难。Cartan推测，但无法证明，他的对合性标准导致了如此有限的延长。Kuranishi证明了Cartan的猜想，尽管有一定的限制。几何分类的现代应用，如Olver和合作者的运动框架方法，导致了复杂的不完全非线性系统。这促使符号算法的发展，包括缺少的约束，包括PI的工作。******不完全系统通常以高指标微分和偏微分代数方程（DAE和PDAE）的形式出现，其中指标是包含缺失约束的微分次数。事实上，这种DAE的复杂性使得计算机在每个阶段都必不可少，从它们的形成，到完成，再到数值解。这促进了强大的问题解决环境的发展，比如MapleSim和SystemModeler。事实上，PI的前学生Wittkopf是MapleSim数值引擎的主要架构师。这样的应用激发了当前的建议，即为包括不等式在内的近似实系统开发补全算法。例如，我们可能需要指定一个未知的密度是真实的和正的，或者机器人手的位置被限制在圆柱体内移动。本建议最重要的部分是建立在半确定规划（SDP）和实际数值代数几何的一些令人兴奋的突破上，以表征这类非线性系统的实际解。具有用户友好界面的计算机程序将使结果广泛可用

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.**