GRK 1632: Experimental and Constructive Algebra

实验与建构代数研究生院的基本思想是在更高的理论水平上对代数问题进行实验研究。在纯数学和应用科学的困难和部分抽象问题的推动下，包括计算机使用的实验方法产生了相关的理论结果和创新的算法方法。RWTH亚琛大学代数研究小组自1968年以来一直在计算机代数方面进行开创性的工作，例如通过开发计算机代数系统GAP、Chevie和CARAT，以及对模块化ATLAS项目的贡献。群论是我们研究的重点；它是普遍存在的对称概念的数学抽象。计算机的作用就像一个显微镜，数学家用它来实验研究抽象的物体。相关方法的不断深入发展，使人们对数学世界的洞察越来越深入。同时，它使其他领域的人们能够使用算法和结果，而不必掌握隐藏在下面的往往困难的理论。正如第一阶段所表明的，这种实验方法为年轻的博士生理解抽象的数学概念提供了一条直接的途径。该研究生院的第二个特点是它的交叉性。参与研究的研究人员在不同的数学领域工作，尽管如此，这些领域仍然有许多联系。这些联系在这所研究生院得到了扩大和加强。由此产生的协同效应产生了创新的方法和不同的观点，导致了算法方法以外的实质性改进。建设性的方法和对称的概念都是共同原则的统一。每篇论文都位于参与教授的至少两个不同研究领域的交叉点，并由至少两名教授积极指导。对不同领域的方法和主题的洞察，确保了博士生的广泛教育，也丰富了导师的科学视角。

The underlying idea of the Graduate School ¿Experimental and Constructive Algebra¿ is the experimental study of algebraic problems at an advanced theoretical level. Motivated by difficult and partly abstract questions stemming from both pure mathematics and applied sciences, an experimental approach including the use of computers yields insight leading to both relevant theoretical results and innovative algorithmic methods.The research group in algebra at RWTH Aachen University has been doing pioneering work in computer algebra since 1968, e.g. through the development of the computer algebra systems GAP, CHEVIE and CARAT, and the contributions to the modular ATLAS project. Group theory is the focus of our research; it is the mathematical abstraction of the ubiquitous concept of symmetry. The computer functions as a microscope the mathematician uses in studying abstract objects experimentally. The continuing further development of relevant methods allows increasingly deep insight into the world of mathematics. At the same time it enables people in other areas to use the algorithms and results without having to master with the often difficult theory that lies beneath. As the first period has shown this experimental approach provides a direct path for the young doctoral students to understanding abstract mathematical concepts.The second characteristic of this Graduate School is its cross cutting nature. The participating researchers work in different areas of mathematics which nonetheless are joined by numerous connections. These connections are expanded and strengthened in this Graduate School. The resulting synergistic effects yield innovative approaches and alternative points of view, leading to substantial improvements beyond the algorithmic methods. Both the constructive approach and the concept of symmetry are unifying common principles. Every dissertation lies at the intersection of at least two different research areas of the participating professors and is actively supervised by at least two professors. The insight into methodologically and thematically different fields ensures a broad education of the doctoral students and also enriches the scientific perspective of the supervisors.