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I'm reading The Princeton companion to Mathematics and Basic Notions of Algebra by Shafarevich. Both of them are really pleasant reading, the first one treats the topics from a more elementary point of view and the second one use advanced examples from other areas and from algebra itself. From this I got to know the Encyclopaedia of Mathematical Sciences of Springer, I'm wondering if you know more texts like this, they don't have to be books, they could be papers/blogs/etc. I'm looking for well explained and motivated introductions to advanced fields/ideas in contemporary Mathematics.

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I think you might find the Unapologetic Mathematician interesting and relevant: – ItsNotObvious Oct 3 '11 at 13:15
  • A book that comes to mind is Abhyankar's Lectures in Algebra. It is a fascinating book.
  • Miles Reid's Undergraduate Commutative Algebra should also fit your description. I am not sure about his other undergraduate book on Algebraic Geometry. I have not read this book.
  • I also like notes by Dr. David R. Wilkins. He has notes on a wide variety of subjects.
  • Notes by James Milne cannot be overlooked.
  • I don't know if Richard Courant's "What is Mathematics?" qualifies as an introduction to advanced areas in contemporary mathemtics. I certainly learned a lot from it at some point.
  • May be, Mumford's Red Book qualifies your criteria?
  • A nice introduction to Algebraic Geometry is Fulton's book "Algebraic Curves".
  • Ravi Vakil's notes on Foundations of Algebraic Geometry are also becoming very popular (though they are not elementary).
  • I also like Cox, Little and O'Shea's "Ideals, Varieties and Algorithms".
  • Rosen and Ireland's "A Classical Introduction to Modern Number Theory"
  • The Catsters on youtube explaining Category Theory.
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Also, the books and notes mentioned above are a bit biased toward Algebra and Algebraic Geometry. I am sure there are great texts for other fields, and would like to know of some. – Rankeya Oct 3 '11 at 8:57

There are a lot of mathematical blogs. For example, I can mention Qiaochu's and also shamelessly plug my own. I can also direct you to look at the "Other Maths Blogs" section of the links on my blog.

Just in case you weren't aware of this: John Baez's This Week's Finds is the granddaddy of mathematical blogging, and contains a lot of gems along what you are looking for.

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