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I would like a suggestion on the 'deepest' books in

Calculus and analysis (something along the lines of Rudin's)

Linear algebra

Abstract algebra

Geometry (and topology); (even something along the lines of New horizons in geometry is acceptable).

Numerical analysis

General physics

Mathematical physics

Probability and statistics

By 'deepest' I mean the ones whose reading can be "useful" for you to help build some kind of mathematical maturity and gain some useful insights. In other words, I'm not asking merely for books which are good for learning a subject (there are lots, I guess), but for some books whose reading can be formative for your mathematical thinking.

Thank you very much.

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    $\begingroup$ I'm guessing that you're using "formative" to refer to development, since you distinguish it from learning. Different views of math education see this separation differently. Therefore, it will be hard to address your question with a book people agree as "formative." There are people who see drilling with exercises as formative, for example. Is there any other characteristic in specific you're looking for? You're most likely just going to get lists of books people personally like, unless you can specify. Although there is nothing necessarily wrong with just getting a list of good books. $\endgroup$ – JPBurke Jul 19 '14 at 14:43
  • $\begingroup$ Good exercises (in particular worked ones) is a good point. Also, I refer to something particularly rigorous and demanding to read yet enlightening and insightful (see Rudin) or simply insightful (see New horizons in geometry). $\endgroup$ – user10024 Jul 19 '14 at 15:00
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    $\begingroup$ I am not sure why "deep" is your concern. Books are written for different audiences. There are books for high school students, undergraduates, graduate students, and researchers. Some books in these areas are conversational in style and others are very formal and technical. Some have lots of examples and others don't. If you specify what level and style of book you are interested in and say a bit more about what your goals are you might get focused answer to your question. $\endgroup$ – Joseph Malkevitch Jul 19 '14 at 15:02
  • $\begingroup$ Is there any particular reason for the inclusion of certain areas in your list and the corresponding omission of others? For instance, general physics is mentioned, but not, say, combinatorics. Also, is the focus supposed to be largely undergraduate or include graduate-level books? $\endgroup$ – J W Jul 19 '14 at 15:19
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    $\begingroup$ @user10024 go to Math.StackExchange e.g. math.stackexchange.com/questions/417167/mathematical-books $\endgroup$ – john mangual Jul 19 '14 at 16:12
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I think of myself as a "geometer" so let me try to offer suggestions for books that I take to be entries to geometry at different levels. Let me first mention that geometry is a huge field and I tried to point this out with my "essay" Geometry in Utopia but I have not updated it in quite some time:

https://www.york.cuny.edu/~malk/utopia.html

What is especially appealing about geometry is that it is so quick starting. One can get to the "research frontier" much more quickly than would be the case for many parts of mathematics.

Some of these books are surveys rather than texts. Once one finds out what areas of geometry one wants to learn more about there are many ways sources to learn more. I want to emphasize that while some of these books are aimed at high school teachers or students that does not mean they can' t be read with profit by graduate students and researchers. Similarly, the books at the graduate school level can also be read by others profitably.

High School:

Coxeter, Harold Scott Macdonald, and Samuel L. Greitzer. Geometry revisited. Washington, DC: Mathematical Association of America, 1967.

Craine, T. and R. Rubenstein (eds.) Understanding Geometry for a Changing World, NCTM, Reston, 2009.

Krause, Eugene F. Taxicab geometry: An adventure in non-Euclidean geometry. Courier Dover Publications, 2012.

O'Rourke, J., How To Fold It, Cambridge University Press, NY, 2011

Undergraduate

Berger, Marcel, and Lester J. Senechal. Geometry revealed: a Jacob's ladder to modern higher geometry. Springer, 2010.

Hartshorne, Robin. Geometry: Euclid and beyond. Springer, 2000.

Pritchard, Chris, ed. The changing shape of geometry: celebrating a century of geometry and geometry teaching. Cambridge University Press, 2003.

Senechal, Marjorie, and George Fleck. "Shaping Space—A Polyhedral Approach." AMC 10 (1988): 12.

Graduate

Goodman, Jacob E., and Joseph O'Rourke, eds. Handbook of discrete and computational geometry. CRC press, 2010.

Grunbaum, Branko, Victor Klee, Micha A. Perles, and Geoffrey Colin Shephard. Convex polytopes. New York: Interscience, 1967.

Grünbaum, Branko, and Geoffrey Colin Shephard. Tilings and patterns. Freeman, 1987.

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  • $\begingroup$ Thank you very much for your suggestions, Professor. $\endgroup$ – user10024 Jul 19 '14 at 17:50
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Take a look at https://math.stackexchange.com/questions/417167/mathematical-books

Math.StackExchange and MathOverflow have a quite a few "book-list" questions

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