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" first got acquainted with Dubrovin/Novikov/Fomenko collection when I was still a second year (sophomore in the US system) student in Math-Phys
trying to learn the basics on plane/space differential curves as a
complement for my Calculus courses. And since then I've made countless references to this book and its siblings.
The pace is fast compared to
other well known introductions on differential geometry/topology
but the text has many insightful and non-trivial examples
examples. Challenging problems are present everywhere in the text!
Though there are a few problems per chapter, the problems sometimes really
require some mastery of the material being far from immediate
applications of the theory developed(for the joy or despair of the
reader). Intentionally the authors try, whenever possible, to replace calculations and
difficult deductions with conceptual proofs. Frequent links with
physical theories (e.g. mechanics, electromagnetism, general
relativity, field theory etc.) compound a good deal of the text
which makes its reading still more delightful. As a con, I would
say the text is quite hard for a beginner but stubborness pays-off in this case.
The second volume of this series covers differential topology w/ emphasis on many aspects of modern physics, like GR, solitons and Yang-Mills theory. There's also a nice account on complex manifolds, mainly Riemman surfaces and it's relation to Abel's thm. Among other topics: classification of compact surfaces , hyperbolic geometry etc.
The third volume covers Homology theory and included a readable account of Spectral sequences for those who may need to learn the machinery for qualifications exams and or applications of complex geometry to contemporary physics (e.g. twistor theory). Viktor Prasolov has recently published a treatise on Homology with more problems and more rigorous proofs. A nice complement to Novikov's exposition.
Eclectic, but at the same time superb. "
Chapter 1. Homology and Cohomology. Computational Recipes
Chapter 2. Critical Points of Smooth Functions and Homology Theory
Chapter 3. Cobordisms and Smooth Structures
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