CAREER: CISE-CCF-AF-Algebra: DMS-Algebra: Computational Differential Algebra

偏微分代数方程（PDAEs）在细胞生物学、化学、数学物理、力学和动力系统、控制理论、微分几何和分析等领域有许多重要的应用。pdae上的算法在简化和解决实际问题方面发挥了重要作用，但随着问题规模的增加，算法的负担也越来越大。效率的提高将减少科学家寻找他们正在研究的系统的特性，也许还有解决方案的时间。该研究包括为PDAEs系统开发一种计算理论，包括那些有参数的系统，并设计算法，提供有效的解决方案描述。对于非参数化系统，PI将首先获得现有微分消除算法的复杂度估计。对于有参数的系统，他将通过表示理论研究其微分伽罗瓦群的结构，这些微分伽罗瓦群是由pdae本身的解组成的线性群。更好地理解它们可以用来提高现有算法的效率或创建新的算法。尽管一个多世纪以来对pdae进行了大量的研究，如Holder、Janet、Riquer、Ritt、Kolchin所开创的，以及Singer、Boulier和Hubert等人最近的进一步研究，但目前还没有足够有效的计算方法来探索和理解这些系统解的微分代数行为。由于提出的研究活动，新的改进的复杂性上界将突出现有算法的一些瓶颈，允许更深入地探索设计更有效的算法。研究活动将进一步发展激进微分理想理论，这对理论和算法发展的任何实质性进展都是必不可少的。该项目将研究的开放问题之一是里特问题，如果解决了这个问题，将为激进的微分理想提供独特的、不冗余的和有效的表示。

Partial differential algebraic equations (PDAEs) have many important applications, such as those in cellular biology, chemistry, mathematical physics, mechanics and dynamical systems, control theory, differential geometry, and analysis. Algorithms on PDAEs have been instrumental in simplifying and solving practical problems but are increasingly taxed as the size of problems increases. Improvements in efficiency will reduce the time for scientists to find properties, and perhaps solutions, to the systems they are working on. The research consists of developing a computational theory for systems of PDAEs, including those with parameters, and to design algorithms that provide efficient descriptions of their solutions. For non-parameterized systems, the PI will first obtain complexity estimates for existing differential elimination algorithms. For systems with parameters, he will study the structure of their differential Galois groups, which are linear groups consisting of solutions to PDAEs themselves, via representation theory. Better understanding of them can then be applied to improve the efficiency of existing algorithms or create new ones. Despite over a century of numerous studies on PDAEs, as pioneered by Holder, Janet, Riquer, Ritt, Kolchin, and recently furthered by Singer, Boulier, and Hubert, among many others, there do not yet exist methods computationally efficient enough to explore and understand the differential algebraic behavior of solutions of these systems. As a result of the proposed research activities, new improved complexity upper bounds will highlight some bottlenecks of existing algorithms, allowing deeper exploration of designs for more efficient algorithms. The research activities will further develop the theory of radical differential ideals which is essential for any substantial progress in theoretical and algorithmic developments. Among the open problems this project will investigate is the Ritt Problem, which if solved, would provide unique, irredundant, and effective representations of radical differential ideals.