Popular Topics in mathematical analysis(Functional analysis)

Question

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.

Answer 1

There is a lot of problems in functional analysis one can mention. For instance,

• The approximation problem. Is every compact operator approximated via finite rank operators? The answer is no in general Banach spaces and yes in Hilbert spaces. (Enflo, 1973)
• Kato conjecture. Are square roots of certain class of elliptic operators analytic? The answer was given is 2002 and you can compare this article.
• Existence and uniqueness of the solution of the Schroedinger equation. Does the Schroedinger equation admit a solution when there is a potential? And is such solution unique? The answer is yes for a large class of potentials, due to the Phillips-Lumer theorem. (1961) However, this is a particularly significative example of a more general problem.
• Existence of topologically complementary of closed sets in Banach spaces. When every closed subset of an infinite dimensional Banach $X$ space has topological complement? When $X$ is Hilbert, thanks to the Lindenstrauss-Tzrafiri theorem. (1970)
• When absolute convergence is equivalent to unconditional convergence? The answer is provided by the Dvrorestky-Rogers theorem (1953) and is: when we are on a finite-dimensional Banach space.
• Self-adjointness of hamiltonians in Quantum Mechanics. Are typical hamiltonian operators of quantum mechanics self-adjoint? Kato-Rellich theorem ensures they are, even for a certain class of singular potentials (such as coulombian ones). (1951)

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:

• Operator algebras ($C^*$-algebras, von Neumann algebras, Banach algebras);
• Nonlinear functional analysis;
• Geometry in Banachs spaces (developed especially from 1970s by Isreaeli school);
• Hilbert manifolds (this topic also developed from 1970s, as far as I know).

Answer 2

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.