Patching in algebra

代数修补

• 批准号: 0901164
• 负责人: David Harbater
• 金额: $ 15.25万
• 依托单位: University of Pennsylvania
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-08-15 至 2013-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0901164&HistoricalAwards=false
• 关键词: Patching; algebra

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.

这个建议涉及代数中修补方法的发展和使用。首席研究员计划使用修补方法来获得新的成果，在理论的二次形式，中央简单代数，司代数，微分代数和伽罗瓦理论，建立在他最近的成果，在这些方向使用修补。 关于二次形式，他将努力推广他最近的结果对维度的各向异性形式定义的功能领域，案件的功能领域或基地领域是更高的层面。 他还将致力于推广他最近的结果期间指数问题的中心简单代数高维的情况下。这将涉及研究和使用地方-全球原则。 在该地区的司代数，他将致力于刻画司代数分裂嵌入问题的解决方案，并延长这一点和他以前的工作在这一领域的情况下，混合和有限的特点。 在微分代数，他将致力于解决分裂嵌入问题的微分伽罗瓦群的特征零，最初在完整的离散估值领域，后来在代数封闭领域和其他大型领域。 他还将研究结构的绝对伽罗瓦集团的领域使用修补，而借鉴他最近修补结果profinite集团是接近免费的。 为了开展这些活动，他将努力进一步扩展他的修补方法，以他最近在领域修补方面的发展为基础，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，长期以来，它们一直被用来通过局部检查空间并观察各部分如何组合在一起来研究空间。 将这种方法引入代数是最近的事，但它使人们有可能解决代数问题，似乎棘手。首席研究员将这种方法引入了伽罗瓦理论，该理论通过检查根的对称性来研究哪些多项式方程可解。 这导致了各种领域的逆伽罗瓦问题的解决方案。 本提案中计划的工作将扩展首席研究员最近的工作，将修补方法转移到代数的其他部分，并解决那里的问题。 他最近开始执行这一计划，获得的结果二次形式，司代数，和其他主题，并提出的工作将超越这一点，扩大适用性的修补方法，并导致结果在几个领域的代数，将超越什么以前可以获得使用其他方法。 通过研究生参加与本提案有关的研讨会和其他活动，本提案的活动还将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群体的成员共同工作，以及通过合作和讲习班加强研究基础设施。