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I am looking for books recommendation on mathematics that is "light hearted", focus on explaining the intuition and thinking process rather than the technical details and rigor.

For example like Pugh's Real Mathematical Analysis, Blitzstein's Introduction to Probability and Marcus's Number Fields which I liked a lot.

I am looking for books like the above on subjects like algebraic geometry, algebraic topology, lie groups, analytic number theory, etc. that are suitable for self study. I understand that those are more advanced subjects and there might not be books like the above but I am gonna ask anyway, thanks.

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A few suggestions:

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Of course this is a list with my personal opinion so what may be too difficult me may be easy for you. Nevertheless, I tried addresing your question on the specific subjects! Here we go :

  • Viktor Prasolov's book " Intuitive Topology" is a good start to get you going on the basic ideas of Algebraic topology.
    • Armstrong's "Group theory" and also "Topology" are really nice books which go a long way to give not only motivation but also some well worked, really non-trivial applications of theory (e.g. finding all finite subgroups of the rigid motions in $\mathbb{R^3}$
    • Munkres' "Topology" is really good and has the reader always in mind pointing out interesting examples and applications or first outlining the more technical parts of theorems.
    • Cristopher Tapp's "Differential Geometry of Curves and Surfaces" has many really illustrative pictures and also whole sections on more advanced or curious theorems ( my favourite is the explanation of how a mechanical contraption called "The south pointig chariot" works.)

Also , AMS has a fairly new (for me atleast) series called Student Mathematical Library, with many excellent books just in the level you describe:Advanced subjects (this means at the final stages of an undergraduate program or the start of a graduate one) done in a way that is both motivated and illuminated by many examples and excersices. Let me mention some of my favourites:

  • Algebraic Geometry: A Problem Solving Approach
  • Introduction to Representation Theory
  • An Introduction to Lie Groups and the Geometry of Homogeneous Spaces (I really like this one)
  • Introduction to topology (you may have heard on of the authors, Vassiliev, on his work on knot invariants, something very trendy right now)
  • e.t.c

Also John Horton Conway's book are a really pleasure to read and study (like "Quartenions and Octonions") but many of them have, for me atleast,missguindgly light-hearted writting while in reality the ideas require a great ammount of effort to be understood.

Finally let me suggest 2 simmilar books, not really in the subjects you ask but really fun and refreshing :

  • "Treks Into Intuitive Geometry: The World of Polygons and Polyhedra" by Akiyama Jin and Kiyoko Matsunaga
  • Apostol's "New Horizons in Geometry"
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Well, that's a tough question, which I think everyone has asked himself, and here's the answer I found by myself (hoping it will be useful for you).

  • Milnor's "Topology from a differential point of view" it's a beautiful and very quick book which introduce (and give some quite deep insight) to the differential topology, with the way of explaining that only a genius such as Milnor can have.
  • Guillemin & Pollack's "Differential topology"
  • Spivak's "A comprehensive introduction to differential geometry". Actually, this is not "a book" is the Bible of the classical geometry, it contains everything you always wanted to know about differential geometry (and, probably, even a bit more), including Lie groups, algebraic and topological geometry. Every definition in this book is explained with an historical background, allowing you to understand how and why this concept has evolved.
  • Weil's "Basic number theory", "Number theory for beginners", "Number Theory: An Approach through History from Hammurapi to Legendre", "Foundations of algebraic geometry". Weil, as one of the great mathematicians of the last century in these fields, writes these books in such a way that oblige you to continously change your point of view. These sort of very active studying might be really hard, but gives you some deep insight, absolutely not trivial
  • Travaglini's "Number Theory, Fourier Analysis and Geometric Discrepancy". Not famous, this book is an excellent mix between analysis and number theory, explaining some concepts that are being investigated these days.

Going off topic, let me suggest another couple of books really interesting:

  • Tao's analysis (one and two)
  • Stein's "Princeton lectures in analysis"
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