Expository texts on advanced subjects

Question

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.

Answer 1

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.

Answer 2

• 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.