The problem of normal forms is to take back some object (matrix, function, vector field) to a simpler one via some group action (linear symplectic or arbitrary coordinate transformations). Heuristically one can imagine change of variables as an infinite dimensional group acting in some infinite dimensional space. The problem is to give this heuristic a mathematical basis.
Moser's original dream was that implicit function theorems will do the job and they have some efficiency in treating the simplest examples. Moser applied this program to simple examples in his two classic papers of 1966 A rapidly convergent iteration method and non-linear partial differential equations.
Hamilton, Sergeraert and Zehnder developed the theory of implicit function to the point it is now known. Deceptively enough, group actions were trated only superficially. In his 1975 classic paper on small denominators, Zehnder ends the first part by a heuristic on group actions, which does not go further than a vague discussion.
Herman took the project to prove the Kolmogorov invariant torus theorem using implicit function theorem. It was exposed by Bost at the Bourbaki seminar but again the group theoretical part remained at an heuristic level and the idea is rather to reduce the statement to the implicit function theorem in Fréchet spaces via a long road of reformulations.
One of the few group theoretic result was given by Sergeraert in his thesis but only for elementary homogeneous actions in differential geometry. In 2010 a similar result was given in analytic geometry by Féjoz and Garay.
So what kind of results are needed for infinite dimensional group actions ? It is reasonable to think that a group theoretic action in infinite dimensional space should include an exponential map. So you should construct a family of operators and a group together with an exponential mapping.
You, reader, may think that you have this already. In any Banach algebra there is an exponential mapping. Well the problem is that a bounded map from a Banach algebra is to specific. Think for instance of the flow of a vector field, there you exponentiate an unbounded operator. If your theory does not include a simple example like that, you will hardly be able to prove any statement about group actions.
First rule: provide an exponential mapping which includes the flow of a vector field as a particular case.
Now perturbation theory involves infinite product on your group which eventually converge to the solution of your problem.
Second rule : provide a criterion for the convergence of infinite products of exponentials.
We were successful in doing this but only in the analytic case, such constructions are unknown in the smooth category.
In the book we used Kolmogorov invariant torus theorem as a starting point but it could have been any theorem on normal forms: hypersurface singularities, vector fields, lagrangian varieties. This is relatively unimportant. There is a total absence of ambition in our book for applications. We do not search any new result but rather to understand thoroughly the classics, so there is actually nothing new in this respect.
In our book on KAM theory we revisited the versal deformation theorem for hypersurface singularities: the Siegel theorem for normal forms of vector fields and the Kolmogorov invariant torus theorem. Note that we do not aim at short proofs at all but rather at conceptual proofs which enlighten the hidden part of original proofs. You can download the book here:
If you are student with a good background on differential calculus and differential geometry wanting to learn KAM theory, this is a good place to start. Certainly you will find in this book, the longest proof of the Kolmogorov invariant theorem that can be found in the literature, but maybe not the least rewarding. It depends very much on what type of mathematician you are. If you think matrices are much simpler than linear algebra, or that using sheaves is a perversion of the human mind, then obviously the books are not for you and you will find many other places to learn KAM theory in a more traditional way.
The reactions of the referees to our book were devastating . You will find below two samples of them. It is a good starting point for a more general thinking on our refereeing system. In this view and for those who speak french, we recommend an interesting video with Laurent Lafforgue where he courageously faces the problem of mathematical communities ruling the academic world.
Reviewer #1: in the preface it is said that this book comes from lectures given around Herman's invariant torus conjecture and the "new language" of functors was chosen in order to "simplify" the exposition and the whole frame where this theory is developed. I know very well the existing literature on KAM theory as well as the notes of Herman himself, transcribed/used or completed by Féjoz, Bost, Fayad and other experts in the field, either in some seminal original papers - where the invariant torus theorem is proved among others - or in some expository review (séminaire Bourbaki etc), and I can say that the language/point of view used in this book is very hard "to digest". I think that it makes things even more complicated than what they already are and I can hardly imagine a dynamicist reading this book in order to understand something about KAM theory, or to learn some clearer and abstract approach to deal with these kind of problems.
Reviewer#2: ...[The book] looks shallow, superficial and ill informed. The authors seem to have taken one or two papers in KAM theory and add a lot of formalism to them. Rather surprisingly, they have ignored many papers by other people [ wrong assertion] Zehnder, Bost, Yoccoz, Broer, Sevryuk, Huitema, Haro, Gonzalez de la Llave, Wagener, Hanssmann, etc. in which the action by groups is considered. I could find very little novelty and not even been up to date of recent developments. The main thing is to develop a notation in terms of category theory."
Beautiful and very neat works already exist on the matter, proposing unified abstract frames in which one can derive several results from a general common functional denominator (the somehow geometrical approach proposed by Herman himself), and I would say that the language/approach of this book kind of betrays the very deep but simple and at the same time proper, "geometrical" and general point of view of Herman' school, from where to look at these conjugacy problems.