The book I would love to read (and own) is called:

**HIGHER MATHEMATICS FOR THE IMPATIENT.** **(HMFTI)**

**Description**: the book appeals to all math aficionados who have at least a solid ground in basic advanced math (say at the level of standard qualifying math department examinations in the USA).

Now, it is sort of similar to the **Princeton Companion of Mathematics**, but without the biographies, and in which the *articles are full-fledged chapters*, with a core introduction (main idea, main results) and a LOT of worked out key examples (take note of the word key, I would like to have only examples which help me grab the essence of the field, and nothing else).

So, just to give you a flavor, take Algebraic K-theory. At some point I kinda knew what it is, but I would love to grab HMFTI, get myself a glass of brandy, a churchwarden pipe, a notebook and a pen, and read the chapter, doing the exercises just enough to have a full sense of the field. Same for the other chapters, say Simplectic Geometry, Orbifolds, Finite Geometry, Higher Categories, etc.

PS This book would be -I think- REALLY nice, but ain't easy to write, in fact extremely hard. You know why? Because it is much much easier writing a text in a field stuffed with all the latest results than a brief introduction which conveys the essentials and nothing but the essentials, all the while being no popularization

reallylike this question... hopefully someone will take a hint and write number (5) and (2) sometime soon! $\endgroup$Hilbert's Problemsbook (in the Proceedings of Symposia in Pure Math series) from the 1970s. Also, Deligne's articleWeil Iis less technical than you might guess, and there is also the textbook by Freitag and Kiehl. $\endgroup$11more comments