Patching in algebra

本提案涉及代数中修补方法的发展和使用。项目负责人计划利用修补方法在二次型理论、中心简单代数、除法代数、微分代数和伽罗瓦理论中获得新的结果，以他最近在这些方向上使用修补的结果为基础。关于二次型，他将把他最近关于在函数场上定义的各向异性形式的维数的结果推广到函数场或基场具有更高维数的情况。他还将把他最近关于中心简单代数的周期指数问题的结果推广到高维情况。这将涉及研究和使用局部-全球原则。在除法代数领域，他将致力于刻画有解的除法代数分裂嵌入问题，并将其与他之前在该领域的工作扩展到混合和有限特征的情况。在微分代数中，他将致力于解决特征零点上微分伽罗瓦群的分裂嵌入问题，首先是在完全离散估值域上，然后是在代数闭域和其他大域上。他还将利用他最近对接近自由的无限群的修补结果，利用修补方法研究绝对伽罗瓦群的结构。为了开展这些活动，他将进一步扩展他的修补方法，以他最近在领域上的修补发展为基础，并试图将这些技术扩展到更高维度的领域。修补方法起源于几何和分析，它们长期以来被用来研究空间，通过局部检查和观察部分如何组合在一起。将这种方法引入代数是最近的事，但它使解决那些看起来难以解决的代数问题成为可能。首席研究员将这种方法引入伽罗瓦理论，伽罗瓦理论通过检查根的对称性来研究哪些多项式方程是可解的。这导致了在各种类型的场上的反伽罗瓦问题的解。本提案中计划的工作将扩展首席研究员最近的工作，将修补方法移植到代数的其他部分，并解决那里的问题。他最近开始执行这个程序，获得了二次型、除法代数和其他主题的结果，并且提议的工作将超出此范围，扩展修补方法的适用性，并导致代数的几个领域的结果，这些结果将超出以前使用其他方法可以获得的结果。这项建议的活动还将通过研究生参加与这项建议有关的研讨会和其他活动，在教育和培训方面产生更广泛的影响。拟议的活动还包括指导和与初级数学家和代表性不足的群体成员共同工作，以及通过合作和讲习班加强研究基础设施。

This proposal concerns the development and use of patching methods in algebra. The Principal Investigator plans to use patching methods to obtain new results in the theories of quadratic forms, central simple algebras, division algebras, differential algebra, and Galois theory, building on his recent results in those directions using patching. Concerning quadratic forms, he will work to generalize his recent results on dimensions of anisotropic forms defined over function fields, to cases where either the function field or the base field is of higher dimension. He will also work to generalize his recent results on the period-index problem for central simple algebras to the higher dimensional case. These will involve the study and use of local-global principles. In the area of division algebras, he will work to characterize the division algebra split embedding problems that have solutions, and to extend this and his previous work in this area to the cases of mixed and finite characteristics. In differential algebra, he will work to solve split embedding problems for differential Galois groups in characteristic zero, initially over complete discrete valuation fields, and afterwards over algebraically closed fields and other large fields. He will also study the structure of absolute Galois groups of fields using patching, while drawing on his recent patching results on profinite groups that are close to being free. In order to carry out these activities, he will work to extend his patching methods further, building on his recent development of patching over fields, and attempting to extend those techniques to higher dimensional fields.Patching methods originated in geometry and analysis, where they have long been used to study spaces by examining them locally and seeing how the parts fit together. The introduction of this approach into algebra is more recent, but has made it possible to solve algebraic problems that had seemed intractable. The Principal Investigator had introduced this method into Galois theory, which studies which polynomial equations are solvable by examining the symmetries of the roots. This led to solutions of the inverse Galois problem over various classes of fields. The work planned in this proposal will extend recent work of the Principal Investigator in carrying over patching methods to other parts of algebra, and solving problems there. He has recently begun carrying out this program, obtaining results on quadratic forms, division algebras, and other topics, and the proposed work will go beyond this, extending the applicability of the patching method and leading to results in several areas of algebra that will go beyond what could previously be obtained using other methods. The activities of this proposal will also have broader impacts in terms of education and training, through the participation of graduate students in seminars and other activities related to this proposal. The proposed activities also involve mentoring and working jointly with junior mathematicians and members of underrepresented groups, as well as enhancing the research infrastructure though collaborations and workshops.