Much of modern mathematics traces its roots to the work of Évariste Galois, whose brief life at the beginning of the nineteenth century brought us the germs of many ideas that now are key in algebra, number theory, geometry, and topology. Two of the areas of active research two centuries later that bear his name are the study of Galois Representations and Galois Field Theory. The former are important tools in number theory that feed into the Langlands Program and many other results, such as the proof of Fermat’s Last Theorem. Meanwhile, the latter is a branch of algebra that has grown out of the so-called Inverse Galois Problem, asking which groups occur as Galois groups of field extensions of a given field. In both cases, people are studying the absolute Galois group of a given field, which can be thought of as the group of automorphisms of the algebraic closure of a field which fix all elements of the field. However, as is often the case with related but distinct areas of mathematics, people working in the two areas do not always work together. A Winter School was organized at the Universite du Luxembourg in February 2012 to bring the two areas together. Several sets of lecture notes from this winter school have been compiled and released in two special issues of Travaux Mathematiques, edited by Sara Arias-de-Reyna, Lior Bary-Soroker, and Gabor Wiese.
The first volume consist of notes from two lecture series given as part of “preparatory days” for the winter school. The bulk of this volume is dedicated to notes on Field Theory by Wulf-Dieter Geyer. Geyer begins with several chapters on the basics of field theory, including a very interesting summary of the history of the concept of fields, as well as the basic facts and theorems. He does not always give full proofs of standard theorems, but instead focuses on developing intuition through many detailed examples. These notes provide a good introduction to the subject appropriate for beginning graduate students. The second half of Geyer’s notes look at more advanced topics, such as the arithmetic of valuations on fields, the Galois theory of pseudo-algebraically closed (PAC) fields, and various theorems on the irreducibility of multivariate polynomials including (but not limited to) Hilbert’s famous theorem which has shown itself to be very useful in studying the inverse Galois problem.
The remainder of the first volume is a set of notes by Luis Ribes on Profinite Groups, which are inverse limits of collections of finite groups and therefore have more structure than arbitrary infinite groups. Given that many books have been written on the subject and Ribes was only giving three lectures, he had his work cut out for him, but his notes do a nice job of summarizing many of the main results in the area while keeping an eye on the connections to Galois theory the whole time.
The second volume deals with the content of the main week of the Winter School, and is divided into five parts. The first part is a survey of the study of Galois Representations written by Gebhard Bockle. Rather than giving a high-level axiomatic treatment of the topic, Bockle chooses to keep things as concrete as possible, writing mostly about the case of elliptic curves. In 30 pages, he manages to write about many deep ideas — Hilbert modular forms, p-adic Hodge theory, the Fontaine-Mazur Conjecture, the Sato-Tate conjecture, and Fermat’s Last Theorem for starters — which he manages to do by keeping the proofs to sketches and referring the reader regularly to sources in his lengthy and detailed bibliography.
There is a pair of articles by David Harbater and Moshe Jarden dealing with the concept of “patching,” which has gotten much attention over the last decade. The idea of patching actually comes from analysis, where one might have two vector bundles over open subsets of a complex manifold that agree on the overlap and combine them to get a vector bundle over the union of the two subsets. New complications arise when one moves to the algebraic setting, including the fact that open subsets are typically dense in a given variety, and people such as Grothendieck, Serre, and Tate have worked on frameworks to extend these ideas. More recently, Harbater and Julia Hartmann have developed a framework of patching over fields that has applications to quadratic forms, division algebras, and the Galois theory of differential equations (among other things), and this framework and its applications is what Harbater discusses in his notes. In his lectures, Jarden looks more specifically of applications of patching to the inverse Galois problem and the theory of PAC fields.
Another set of notes is by Michael Schein and is about Serre’s Modularity Conjecture. He gives a survey of what the conjecture says (roughly: Any odd irreducible two-dimensional Galois representation over a finite field arises from a modular eigenform and we know what the weight and level of the modular form should be) as well as a proof and some generalizations. The volume concludes with a research article by Geyer and Jarden which gives a self-contained presentation of a result of János Kollár on the density property of PAC fields and then uses this result to prove that the theory of PAC valued fields is model complete.
Most of the writing in these volumes is expository and can be found in other sources. However, the articles contained in here are all well-written and it is nice to have them all contained in one place with detailed bibliographies. For a graduate student interested in learning about what Galois theory looks like in the early 21st century, I think these books make a good starting point.
Darren Glass is an Associate Professor of Mathematics at Gettysburg College. He was trained as a Galois Theorist, although in recent years his research has shifted towards combinatorial number theory and graph theory. He can be reached at email@example.com.