This book is worth reading for its second (stand-alone) half, about topology and physics. Michael Monastyrsky (MM) relies mainly on words to convey the big picture of many connections between the two topics. I had a number of "aha" moments while reading it, which had eluded me despite having struggled through some more formal introductions to T&P. Chapter 9's discussion of the relationship between homology and homotopy and later chapters about liquid crystals and topological particles were especially enlightening.



That said, I think it would be difficult to have those moments if you hadn't already had at least an introduction to algebraic topology (e.g., Michael Henle's wonderfully clear "A Combinatorial Introduction to Topology," from Dover). The treatment gets more abstruse in the later chapters, where it feels like MM was rushing. E.g., the chapter on braids and knots, which should be relatively intuitive to understand, struck me as quite abstract even though I'd already read a couple of books on the subject. And the discussions of magnetohydrodynamics and "What's next?" (at the end) were more like catalogues of topics than anything you could learn from. MM is also sometimes quite rambling, such as in his long chapter on gauge fields, in more than 80% of which he doesn't discuss topology at all.



The rambling tendency is even more evident in the Riemann part of the book. E.g. Dirichlet doesn't make an appearance in the chapter entitled "Riemann and Dirichlet" until, again, 80% of the way through the chapter; and in the 2-page chapter "Last Years", MM jumps from Riemann's marriage and illness in 1862 to June 28, 1866, then to early 1866, then to June 14 and finally to Riemann's death on July 20 (@75-76). Even though Riemann's accomplishments are extremely interesting, there's something flat and dull about the writing style in this part of the book, so much so that I'd lost interest on two previous attempts to read it. But the main flaw in this section is that MM seems unsure of what level of readership he's aiming for. E.g., on one hand MM feels readers need to be told that "shock waves are formed when high-speed aircraft break the sound barrier, when atomic bombs explode, and so forth" (@69), but on the other hand if you don't have any prior background in complex analysis you will be lost. (It also doesn't hurt to have encountered monodromy mappings previously (@56)).



A few more diagrams would have been helpful especially in the Riemann section, and throughout the book it would have been nice if the diagrams had been re-drawn from the Russian edition: some of them are quite murky, and one or two of them don't seem to match the revised text. All in all, I give "Riemann" barely a 3, and "T&P" a 4, for an average of 3.5 stars.

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