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What are the latest major discoveries in math? What areas and problems are considered important for overall advance of math?

Also, given that most of those topics are most likely very advanced - what is the best learning path to approach them?

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closed as not constructive by Noldorin, Casebash, Jamie Banks, Simon Nickerson, Peter Smit Jul 21 '10 at 7:55

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This is far too wide and vague to be an answerable question in my mind. –  Noldorin Jul 21 '10 at 7:41
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voting to close, but I would not object to certain reformulations of this question as long as they were limited to a much smaller domain. –  Jamie Banks Jul 21 '10 at 7:47
    
Agreed with Katie... this could be a viable question, but only if reformulated and reduced to a much narrower scope. –  Noldorin Jul 21 '10 at 8:01
    
This is a constructive question in my opinion. This was an attempt to learn. I searched and found this very topic after 3 years it was asked. and i found one of the answers to be useful. up-voting that answer as well. –  Rakesh Jul 15 at 5:07

2 Answers 2

There are a few places on the web that have lists of open problems in specific subfields. For example,

The Open Problems Garden

(this is not an endorsement, I don't know much about the site besides its existence)

and

The Open Problems Project

which is specific to discrete and computational geometry. The latter is a favorite of some people looking for nice research projects for undergraduates or high school students, as the problems tend to have statements, at least, accessible without too much background.

The broader question you ask is one I won't address, by dint of intractability.

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Higher category theory and Brave new algebraic geometry.

If you want to approach these subjects, you should take a look at (roughly in this order):

Algebra by Lang

Categories for the working mathematician by Mac Lane

Categories and Sheaves by Kashiwara-Schapira

Basic Concepts of Enriched Category Theory by Kelly

Model Categories by Hovey

Model Categories and their Localizations by Hirschhorn

Simplicial Homotopy theory by Goerss-Jardine

Notes on Quasicategories by Joyal

Higher Topos Theory by Lurie

Higher Operads, Higher Categories by Leinster

Introduction to commutative algebra by Atiyah-MacDonald

Math 614 notes by Hochster

Math 615 notes (2010) by Hochster

Derived Algebraic Geometry (thesis) by Lurie

Homotopical Algebraic Geometry (I&II) by Toen-Vezzosi

In all seriousness though, this question is stupid.

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^this comment is stupid. –  The Chaz 2.0 Oct 19 '11 at 21:10

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