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.
Best Answer
There is a lot of problems in functional analysis one can mention. For instance,
A never closed (so far) problem is that of Navier-Stokes equations, in the precise statement of the 6-th Millennium Problem. The 5-th Millennium Problem, concerning Yang-Mills theories, would probably be strictly connected with functional analysis in its solution. Furthermore, I remember my lecturer said Perelman proved Poincaré conjecture (more precisely, Thurston' geometrization conjecture) making use of functional analysis methods (2001-2002).
Obviously, such a list is subjective, in the sense those problems are the ones has impressed me so far and surely is incomplete. Moreover, they reflect my own formation. I must point out I don't really know how all of those problems were popular at the time they were open, and that some of them are rather specific, but I think they should be mentioned at least in view of the importance of their applications.
Added. For the third part, I'd say: