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.
