Patching methods in algebra were motivated by ideas that originally arose in analysis, beginning with work of Riemann and Hilbert. Later work on this theme was done by others such as Birkhoff, Zariski, Cartan, and Serre. This work enabled the construction of global objects from more local objects.
Later results of Grothendieck and of Tate made it possible to use these ideas in studying Galois groups and étale fundamental groups, beginning with work of the speaker in the 1980's. Much more recent work (e.g. [HHK09], [HHK11]) has applied patching to a wider range of algebraic objects, such as quadratic forms, division algebras, differential modules, and torsors, over certain types of function fields. This work relies on a new approach to patching that requires little machinery, while still drawing on the basic geometry of algebraic curves (see [HH10]).
Among the applications of patching originally considered by Riemann and Hilbert was the problem of realizing matrix groups as monodromy groups of linear differential equations. Recent results carry this over to a more algebraic setting, permitting the realization of linear algebraic groups as differential Galois groups, and making it possible to solve differential embedding problems, over more general function fields in characteristic zero.
A non-commutative analog of the inverse Galois problem was introduced by Schacher, under the term ``admissibility''. The question is whether a given finite group G is the Galois group of some extension E of a given field F such that E is a maximal subfield of an F-division algebra. Various partial results have been proven for global fields F. Patching results have made it possible to obtain results over other function fields, including necessary and sufficient conditions for admissibility in certain cases.
Other recent patching results have concerned the u-invariant of a field F, which is the maximal dimension of an anisotropic quadratic form over F; and the period-index problem for F, which relates the order of a class in the Brauer group to the degree of the division algebra in that class. These results arise from local-global principles that in turn can be proven using the new patching methods. Those principles make it possible to obtain results about given global objects by studying these objects locally, and can be viewed as generalizations of the classical theorems of Hasse-Minkowki and Brauer-Hasse-Noether.
This mini-course will begin with a brief overview of patching methods in general, and will then describe the new approach to patching which was developed in recent years by the speaker and Julia Hartmann. The course will then discuss applications of this approach, especially those obtained in joint work of the speaker, Julia Hartmann, and Daniel Krashen. This includes topics mentioned above, as well as other applications such as patching torsors.
References
Last modification: 28 October 2011.