Is there any Book/Book Series/Website which illustrate advanced mathematics but in concise and basic form. I just want to be familiar with the literature and I don't want a deep exploration of the topics. What I want is a near complete survey on different topics of math, What is important for me is the basic ideas and the connections between different parts of mathematics. For example if it talks about abstract algebra, then a quick survey of the concepts of group, ring,... and how these are connected to other topics...I just want to get a obscure grasp but pretty comprehensive topics.

BTW, I know there might not any reference but there might be a series of books. Or different book for different topics e.g. one of Geometric Algebra one for stochastic processing and so on. You know, I just want to get a good grasp of literature as a whole.

p.s. currently I'm interested in functional analysis, harmonic analysis, number thoery and operator theory.

  • $\begingroup$ This is a very broad question, but Wikipedia's article on mathematics (en.wikipedia.org/wiki/Mathematics) includes a surprisingly good survey. $\endgroup$
    – Mr. Chip
    Mar 7, 2013 at 13:28
  • $\begingroup$ Thanks Joshua, Yes, you're right and this is very broad. But I am not a math student and I also have some primary knowelege of undergraduate level. But I like to get a glimpse or vision of math. What is a ring? what operator theory primarily and fundamentally saying? what is the connection between geometric algebra and discrete geometry?! I just want a introductory level of information broadly (but in a quick way:( ). $\endgroup$
    – SddS
    Mar 7, 2013 at 13:39

2 Answers 2


Luckily for you, one such book was compiled fairly recently: The Princeton Companion to Mathematics.

The first part of the book contains an overview of modern pure mathematics which should allow you to hit the ground running. The third part contains a survey of 99 important mathematical concepts, and the fourth part contains longer surveys of 26 major branches of mathematics. There are copious cross references indicating how different branches and different concepts cross-pollinate.

You can see the table of contents here yourself.

Added later: actually, a second such book has also been compiled: The Princeton Companion to Applied Mathematics. It's of a similar layout to the "pure" Companion but covers very different topics. Some more information on the book, including a pdf of the table of contents, is provided at the linked webpage.

  • $\begingroup$ Thank you Willie for your great suggestion. $\endgroup$
    – SddS
    Mar 19, 2013 at 17:47
  • $\begingroup$ It might also be worth explicitly mentioning The (less well-known?) Princeton Companion to Applied Mathematics. $\endgroup$
    – Will R
    Apr 28, 2016 at 12:46
  • $\begingroup$ @WillR: I agree it should be mentioned. I didn't mention it before because it was published after my answer was written. Feel free to edit it into the answer! $\endgroup$ Apr 28, 2016 at 13:04

OK, I do a search and sth found (mostly as Joshua said from available from Wikipedia reference part).

Encyclopedia of Mathematics This is good website but search based and a bit far from what i wanted but quite handy. I don't know how much it is comprehensive in the matter of subjects and how much efficient in showing the relation,...

HyperMath Far from what I needed, it's quite basic and incomplete.

FreeScience This is collection of free resources. Nothing more!

The Math ATLAS Pretty close to what I need but seems incomplete and obsolete. The good thing about this website is it has classified the literature pretty close to what I need. The best classification approach. It also cite some text and references for each topic which is cool. but I need a book/website with discussing the topics (pretty concise), make a relation with other topics and a bit uptodate.

Online Mathematics Textbooks this also provides some free books on different subjects.

Tricki Tries to collect different problem solving approaches! It also have concise desecration of each topic. pretty incomplete.... How much successful? I don't know. (here is the site's map)

PlanetMath To me not a good choice...I didn't like its topic classification approach either (here)

Math Structure a comprehensive list of mathematical structures. Beautiful idea! (Abstract Algebra)

Wolfram Pretty basic and totally fragmented.


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