[Math] Popular Topics in mathematical analysis(Functional analysis)

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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,

  • 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).
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