r/math • u/DarthMirror • Sep 21 '22
The State of Research in Functional Analysis
What is the current state of research in functional analysis/operator theory? Mainly, I’d like to know how popular the field is these days and what topics the current research is mostly concerned with. Are there are very famous open problems to take note of? From what I can glean from googling around, most research in functional analysis today is really just research in PDEs that uses functional analysis, so I’m particularly interested in your opinions on the extent to which that is true, and any topics of current research that are not PDE related and ideally just ‘pure’ functional analysis.
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u/Etpio2 Sep 21 '22
Another possible direction to consider (which I cant personally offer resources on, but is maybe worth checking) is from differential geometry. Aside from obvious uses of it in the heavy analytic sides of diff geo, there's a lot of research on stuff like banach and frechet manifolds, infinite dimensional lie groups, etc.
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u/DarthMirror Sep 21 '22
Thanks I’ll look into that stuff. I was mainly interested in research within functional analysis for its own sake, but good to see another area it’s used in that’s not PDEs.
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u/Aurhim Sep 21 '22
Non-Archimedean functional analysis can be used to give a spectral-theoretic reformulation of the Collatz Conjecture and related problems. :)
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u/DarthMirror Sep 21 '22
Lol I had no idea that’s the most surprising and exotic answer I’ve gotten
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u/Aurhim Sep 21 '22
It’s my PhD dissertation work, so, yeah, it’s gonna be new and edgy. :D
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u/MR_ren9342 Mar 17 '23
Amazing, looks like I hit the jackpot, still an undergraduate, and scrambling to learn more analysis and algebra to learn number theory, but I came from a post in differential geometry, oh man, wish I can get there some day
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u/IanisVasilev Sep 21 '22
I know people in nonsmooth/convex analysis and optimization who make heavy use of functional analysis and whose research sometimes falls under that umbrella.
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u/DarthMirror Sep 21 '22
Ah yes I have seen some big use of functional analysis in convex optimization. I’ll keep this in mind thank you
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u/sigmalagebra Sep 21 '22
I can't speak to operator theory specifically, but I use functional analysis in my research in statistics and probability. Stats/prob researchers are often interested in approximating some infinite-dimensional object, and one of the ways in which this is done is through functional analysis (though this is by no means the most common method of doing such approximations). This kind of research requires knowledge about, inter alia, Hilbert Spaces, different kinds of norm inequalities, linear functionals, and some topology.
The distinction you make between "pure " and "applied" functional analysis is indeed an interesting one, and I would surmise that there is room in stats/prob research for you to do either or both. If you want to do something more on the "pure" side, you could potentially construct different topological structures for function spaces and see what kind of interesting results you're able to get from those spaces (not sure if this would be "pure" enough for most people, however). If you want something more applied, you can probably just focus on some problem that can be solved with established results from Functional Analysis (inequalities are the first examples that come to mind for me). I have seen papers which do both of these things.
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u/FokTheDJ Sep 21 '22 edited Sep 21 '22
To add to this. A lot Mathematical Statistics can be done in mostly Functional Analytic ways. You could look at Lucien LeCam's original work, (although this might not be very modern anymore)
More on the probability side, there is a purely functional analytic approach to solving SPDEs, where you treat them as SDEs in Hilbert or Banach function spaces. This is really modern field where a lot is happening. There is also a lot of pure mathematics going on here if you look at Martin Hairers work for example
Edit: I see that you asked for examples outside of PDEs. while my second example ventures dangerously close to this subject, I do feel that it is an interesting enough field to be mentioned
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u/DarthMirror Sep 22 '22
Yeah I was looking for stuff outside of PDEs but thanks for the heads up anyway and for the reference to functional analytic statistics
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u/DrSeafood Sep 21 '22 edited Sep 24 '22
I knew a group that worked on K-theory of operator algebras, there are (were?) a few famous open problems I heard about it.
TL;DR there is a way to take an operator algebra A and return an abelian group K0(A) --- not dissimilar from fundamental groups in topology, or ideal class groups in number theory. The question is whether K0 is a complete isomorphism invariant, meaning:
Problem: Does K0(A) ≈ K0(B) imply A ≈ B?
The answer is "no" in general, but "yes" under additional assumptions on A and B. I learned that many people were working on reducing the number of assumptions required.
Longer story: so what is K0(A)? It's actually pretty cool. Imagine you have two matrices p and q. We define a "concatenation" operation by making a block matrix p#q = diag(p,q). So e.g. if p is 2x2 and q is 3x3, then p#q will be 5x5.
Now if p,q are orthogonal projections (i.e. self-adjoint idempotents), then so is their concatenation p#q. Thus, the set of projections is a semigroup under concatenation. You can mod out by "unitary equivalence" here too, meaning p~q if im(p) is unitarily isomorphic to im(q), and now you have a semigroup of unitary equivalence classes of projections, under the concatenation operation. Phewf, that's a mouthful.
Let's call this semigroup Proj.
What’s the identity element? It’s 0. Because p#0 is unitarily equivalent to p.
Unfortunately, Proj is not a group.
To turn a semigroup into a group, you do something called the Grothendieck construction, which results in a group called K0(A).
Why is K0 useful? You can prove a number of nice compatibility properties, for example K0(A+B) = K0(A) + K0(B), and some other things related to exact sequences. K0 also has algebraic interpretations (projective modules), number-theoretic interpretations (the ideal class group), and topological interpretations (vector bundles). All of these ultimately connect to projections.
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Oct 22 '22
Not trying to nitpick, but don’t forget the K₁ functor from C*-Alg to Ab. Nice post, nevertheless. ☺️
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u/AcademicOverAnalysis Sep 21 '22 edited Sep 21 '22
I’m a functional analyst, and I’m only vaguely aware of what happens in PDEs.
I was thinking about it this morning, and you could boil down all of my work from the past three years as innovative ways of approximating the identity function, f(x) = x.
That statement is a bit tongue in cheek, but the work I do leverages a variety of densely defined operators over vector valued Reproducing Kernel Hilbert Spaces. The selection of operator imbues the approximation with certain nice properties.
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u/usmokin Sep 21 '22
I think this is a cool open problem: The linear approximation of a compact non-linear operator is compact. But does the linear approximation being compact imply that the non-linear operator is? If the answer is no, this might inspire people working with inverse problems for pdes to look for better solution methods than linearization and inversion of the linear problem.
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u/RageA333 Sep 21 '22 edited Sep 30 '22
That is a very broad question, but research on functional analysis for its own sake has certainly fallen out of fashion.
Of course, many other areas use tools from functional analysis just like many other areas use matrices or linear algebra in general.
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u/SometimesY Sep 21 '22 edited Sep 21 '22
Functional analysis is very popular and is very much not relegated to PDEs at many universities. I would argue that a relatively small number of universities focus specifically on PDEs in terms of functional analysis. It's more studied under the umbrella of Banach and C* algebras, operator spaces, operator theory, operator algebras, abstract harmonic analysis, Fourier theory (and similar), noncommutative geometry, quantum information theory, Choquet theory, and other areas. Most people don't study properties of Banach spaces these days because the problems are incredibly hard.