TITLE: Popular Topics in mathematical analysis(Functional analysis)
QUESTION [6 upvotes]: I am writing a text(as a duty by my mentor) dealing with the recently popular topics(including open problems) in mathematical analysis. At first part, I briefly introduced the mathematical analysis(and functioanal analysis) and gave the sub-branches (like real, complex and numerical analysis etc.) it includes.
At  second part I mentioned about open problems(Hilbert's Problems, Millenium problems etc.) in mathematical analysis.
At third part I plan to mention about recently popular topics(may be from the date of 1900).  
For  the informations in part I and Part II, I can find objective criterias and official sites(wikipedia etc.) so I can support whatever I wrote by citing these sites. But for third part I dont know how to reach such an information.(In fact topic "popular" is subjective) First topics come to my mind are fuzzy set theory(1965), theory of set-valued functions(1950). Could you suggest more topics which are popular  recently?
Thanks for your helps.

REPLY [2 votes]: A recently solved (last summer) conjecture is the Kadinson Singer Conjecture. http://gilkalai.wordpress.com/2013/06/19/the-kadison-singer-conjecture-has-beed-proved-by-adam-marcus-dan-spielman-and-nikhil-srivastava/
This has application relevant to quantum mechanics.  It was proved via the Paving Conjecture, which is equivalent.  The solution doesn't use super advanced techniques, but does take advantage of spectral theory.