I'm a physicist trying to gain a deep understanding of mathematics that is required for my work.I intend to specialize in string theory which is a very math intensive branch of theoretical physics that requires knowledge of algebraic geometry , Algebraic topology and virtually every other branch of math.I noticed that most mathematicians focus on a very narrow field of research for most of their careers. Here's a problem. I want to gain a deep understanding of most branches used in string theory research to the point of being able to use the methods of , For example, Algebraic geometry to solve problems that arise in my research in physics . Is there a quick way to do ? There doesn't exist a textbook of algebraic geometry for physicists besides modern research use methods in algebraic geometry , Combinatorics etc that are unknown to most physicists. These tools are present only in pure math books and math research papers

19$\begingroup$ If your question is "Is there a quick way to gain a deep understanding of most branches of mathematics used in string theory?", the answer is most certainly no. Big things take a lot of time. $\endgroup$ – Joonas Ilmavirta Feb 4 '15 at 7:37

9$\begingroup$ This is very applicable here. $\endgroup$ – Asaf Karagila Feb 4 '15 at 8:13

7$\begingroup$ I suspect you would get much more helpful answers to this question from physicists than from mathematicians. $\endgroup$ – Hjalmar Rosengren Feb 4 '15 at 8:35

6$\begingroup$ physics.stackexchange.com/questions/2528/… $\endgroup$ – Carlo Beenakker Feb 4 '15 at 9:45

8$\begingroup$ Most mathematicians focus on a very narrow field for a reason: it takes all their time to get a good (maybe even deep) understanding of that narrow field. I am tempted to say that you are asking “the answer to life the universe and everything” in a quick way. Well, the quick answer is: 42. $\endgroup$ – jmc Feb 4 '15 at 9:48
The book "Mirror Symmetry" by Hori, et.al. (Clay Mathematics Monographs) contains a sort of crash course on the primary subjects of geometry and topology needed for string theory. It is "Part 1" of that book and compresses into about 150 pages what would normally be covered by several books or several graduate math classes. It was written by Eric Zaslow with exactly people like you in mind  physicists trying to come to grips with the mathematics required for string theory (there is a parallel crash course in physics for mathematicians in that book  "Part 2"). Part 1 is necessarily terse and is sort of a like Cliff Notes for geometry  you can't really learn the subject from it, but it can be a great place to start and might be good enough for many of your purposes. The book was written around the year 2000 which is pretty ancient in stringtheory years, but the Part 1 holds up pretty well. If it were rewritten today, it might include a few more topics like derived categories and perverse sheaves, but it is a good start for what you want.

$\begingroup$ actually I just noticed that chapter 38 of that book (written by Richard Thomas) includes an introduction to derived categories in the context of Dbranes. $\endgroup$ – Jim Bryan Feb 5 '15 at 0:15
On this website http://www.theorie.physik.unimuenchen.de/activities/schools/archiv/2011_asc_school/index.html there are videos of the lectures given at the 2011 Arnold Sommerfeld PhD School ''Algebraic Geometry for String Theorists''. For each lecture you will find also reccomended references for "Further reading".
This textbook "Algebraic Geometry: A First Course", by Joe Harris (BookZZ has an electronic version of it) is not particularly oriented at phisics applications but provides an introduction into the subject "with an absolute minimum of technical machinery". See also these lectures http://www.mathematik.unikl.de/~gathmann/class/alggeom2002/main.pdf by Andreas Gathmann from which I have learned about the Harris' book.
For algebraic topology, the survey article Algebraic topology geared towards physicists could be helpful (cf. also e.g. this Physics.SX question).