This is a rather pretentious title, as the author remarks in the introduction, but as we shall see there is some justification for this. First its contents.
There are 3 chapters, altogether 309 pages and 3 appendices on functional analysis and differential geometry. Chapter 1 discusses classical ideas and problems like max-min principles, WKBJ, Rayleigh-Ritz, spectral theory. One is introduced to classical mechanics (8 pages) and quantum mechanics (again 8 pages). It is full of remarks like “another version of the theorem was proved by..., we mention also its common background with... and one should note its fruitful interaction with... in the field of...”. Chapter 2 is called scattering theory and solitons. It is a comprehensive account of operator theory, spectral methods and integrable systems. Chapter 3: Some nonlinear analysis: some geometric formalism introduces topological methods, semigroup theory, variational inequalities, quantum field theory, gauge fields.
Although the author admits some earlier flirtation with Bourbaki, the style is acceptable for mathematicians of applied denomination and the text is accessible to physicists. On the other hand, as mentioned above, the book seeks to outline relations and common ideas to various theories, which presupposes a lot in background knowledge; also because of the large amount of material presented the discussion of all kind of concepts is necessarily restricted. I checked this by choosing 4 items from the index: the Huygens principle, Higgs field, the Brouwer fixed point theorem, the Ginzburg-Landau equation. For the Huygens principle one finds only that a certain formula represents this principle, for Higgs field one finds again an indication of a relation with something else. The Brouwer fixed point theorem gets some more space: a proof of 3 lines !, an indication of possible application and another proof of 3 lines (this proof uses degree theory and actually needs many pages). The Ginzburg-Landau equation(s) crops up in several contexts but not in its usual form; also its increasing importance is not made clear.
Still I feel that this book is a very rich source of material for experienced mathematical physicists. If one wishes to adopt it for a seminar I think one can have a lively time but I strongly advise to hire the author at the same time to supply additional information.