The theory of distributions is very interesting, and I have noticed that it has many applications especially with regard to PDEs. But what are the research topics in this theory? also in terms of functional analysis

4$\begingroup$ just a quick link to this related question I posted a while ago mathoverflow.net/questions/259834/… $\endgroup$ – Abdelmalek Abdesselam Apr 6 '18 at 17:07

2$\begingroup$ These days distributions are used in almost any area of PDEs in the sense that weak derivatives are used routinely. $\endgroup$ – Deane Yang Apr 7 '18 at 0:35

3$\begingroup$ There is also stochastic PDEs  the white noise term is distribution valued. $\endgroup$ – user69208 Apr 7 '18 at 1:07

2$\begingroup$ The mathematical foundations of signal analysis and of quantum field theory, just notifying of two examples that I went through. $\endgroup$ – Alan May 8 '18 at 15:49
While I do not know much about current development of the general theory of distributions, I can say something about the current research topics in a special class of distributions, the theory of Sobolev spaces.
Theory of Sobolev spaces was one of the greatest discoveries in the XXth century mathematics. This theory is the most important single tool in studying nonlinear partial differential equations, both in its theoretical aspects and numerical implementation. Although the theory of Sobolev spaces has been created in the late thirties, in recent years, there have been major breakthroughs in the theory, by expanding the applications to new areas of pure mathematics like analysis on metric spaces, geometric group theory or algebraic topology as well as to areas in applied mathematics, like for example to nonconvex calculus of variations.
I will list some of the active research areas, just to set an example, but the list is far from being complete. I will focus mostly on the areas that I am familiar with. These areas are not directly related to partial differnetial equations, where the apllications are well known. For each topic I will provide just one reference as otherwise I would have to put hundreds. It will then be easy to search MathSciNet to find other relevant references.
 Sobolev spaces on irregular domains. The classical embedding and extension theorems assume that the boundary of a domain is quite regular. However, a substantial effort has been put in extending these results to domains whose boundary might be fractal. It seems that the first important paper in that direction was [6].
 Sobolev mappings between manifolds. This class of mappings appear in a natural way in the study of geometric variational problems for mappings between manifolds. Like for example, the theory of harmonic mappings. One of the early problems was the question whether smooth mappings are dense. That led to a very fruitful research showing deep connections to algebraic topology. See for example [4].
 Theory of quasiconformal and quasiregular mappings. Quasiconformal mappings are homeomorphisms that distort balls in a certain controlled way. Quasiregular mappings are roughly speaking quasiconformal mappings that can have branching set where they are not onetoone. Just like conformal mappings versus holomorphic functions. One of the main tools in study of such mappings is the theory of Sobolev spaces. Applications of the theory include conformal parametrizations of surfaces, dynamical systems and rigidity results like the Mostow rigidity theorem. For a recent impressive result see [3].
 Mappings of finite distortion. The theory of quasiregular maps led to this theory. It has important applications in the nonlinear elasticity. For basic results, see [7].
 Nonlinear elasticity. An approach to nonlinear elasticity proposed by J.Ball [1], led to new questions in the analysis and geometry of Sobolev mappings. This development is related to the theory of quasiregular mappings and mappings of finite distortion discussed above.
 Variable exponent Sobolev spaces. This is the extension of the theory to the case in which the $L^p$ spaces are replaced with $L^{p(x)}$. That is, the exponent of integrability is a function [2].
 At last, but not least: Analysis on metric spaces. Quite surprisingly, the first order analysis involving derivatives can be extended to metric spaces equipped with a measure. This is a very active research area that has already its MSC classification 30L. Sobolev spaces on metric spaces play an important role in the development of this theory [5].
[1] J. M. Ball, Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal. 63 (1976/77), 337–403.
[2] L. Diening, P. Harjulehto, P. Hästö, M. Růžička, Lebesgue and Sobolev spaces with variable exponents. Lecture Notes in Mathematics, 2017. Springer, Heidelberg, 2011.
[3] D. Drasin, P. Pankka, Sharpness of Rickman's Picard theorem in all dimensions. Acta Math. 214 (2015), 209–306.
[4] F. Hang, F. Lin, Topology of Sobolev mappings. II. Acta Math. 191 (2003), 55–107.
[5] J. Heinonen, P. Koskela, N. Shanmugalingam, J. T. Tyson, Sobolev spaces on metric measure spaces. An approach based on upper gradients. New Mathematical Monographs, 27. Cambridge University Press, Cambridge, 2015.
[6] P. W. Jones, Quasiconformal mappings and extendability of functions in Sobolev spaces. Acta Math. 147 (1981), 71–88.
[7] S. Hencl, Stanislav; P. Koskela, Lectures on mappings of finite distortion. Lecture Notes in Mathematics, 2096. Springer, Cham, 2014.

$\begingroup$ Great answer. I always feel I need to learn Sobolev spaces again. $\endgroup$ – Bombyx mori Apr 6 '18 at 23:25

$\begingroup$ Maybe you should split this up into several answers? That way we can like each one. $\endgroup$ – Deane Yang Apr 7 '18 at 0:31

2$\begingroup$ @DeaneYang I think I prefer to keep all examples in one place. It is easier to read. Otherwise it would look like trying to get more reputation for one answer :) $\endgroup$ – Piotr Hajlasz Apr 7 '18 at 0:35

$\begingroup$ @PiotrHajlasz I wanted to wait a while before accepting your answer, but it is certainly the most satisfying: thanks for your contribution. $\endgroup$ – Andrew May 28 '19 at 20:42
I think this is a good question. I am no longer working in this field. When I was working in this field, there is no general program and I do not know what are the important open problems. Instead of following the useless advice "ask your advisor", I wish I had asked this when I was a young graduate student.
I would characterize the situation for distribution theory and linear operators as "algebraic vs analytic". On the algebraic side, you can delve into nuclear spaces, nuclear operators, noncommutative geometry, deformation quantization, microlocal sheaves, and maybe a lot of other related topics in representation theory. On the analytic side, you can delve into scattering theory, elliptic PDE, Hodge theory on noncompact manifolds, and other problems arise from mathematical physics. There is some interaction between the two sides, but my impression is that people on one side does not necessarily know the machinery used in the other side.
Perhaps a better question to ask is "What problems in (other fields of mathematics) I want to solve using microlocal analysis?". I think the theory of linear operators, like the language of $\epsilon\delta$, is ultimately valuable only if you know how to make use of it to build other things. Personally I am excited with the fact that $\det(\Delta)$ is related to partition functions in quantum field theory and has an interpretation in topology. I have never seen any one investigating how to regularize$\sum_{\lambda_{i,j}\in Spec(\Delta)} \lambda_{i}\lambda_{j}$, and one reason may be there is no obvious association to other fields. I imagine you will be interested in a lot of other subjects as well. If you found some topic exciting and you can attack it using the machinery you know, I think this may be a decent research problem already.

$\begingroup$ Do you know any other details about elliptic PDE? $\endgroup$ – Andrew Apr 6 '18 at 18:01

1$\begingroup$ I do not know. I have seen papers on this many years ago, but my knowledge is definitely out of date. If you are into elliptic theory, check work of Juncheng Wei and Yanyan Li. $\endgroup$ – Bombyx mori Apr 6 '18 at 18:27
I Don't have time for a very elaborate answer (will expand later), but I think the main research questions about Schwartz distribution relate to probability theory on spaces of such distributions. This is basically what quantum field theory is about. Some of the recent developments relate to probability measures on $\mathcal{S}'(\mathbb{R}^d)$ obtained as scaling limits of discrete spin systems.
See for example:
https://arxiv.org/abs/1803.03044
and

$\begingroup$ Maybe I will get downvote for this comment. But I wonder why you make efforts with "probability theory" to explain physics. In my opinion, quantum mechanics is formulated badly.(Obviously this is offtopic, in this context) $\endgroup$ – Andrew Apr 7 '18 at 0:30

4$\begingroup$ @Andrew: I am not sure I understand your comment. Are you referring to the old controversy about the Copenhagen interpretation etc.? I stay away from that kind of thing. I am happy letting physicists debate among themselves about how QM should formulated. As a mathematician, I simply take their consensus theory and try to work out the mathematical aspects. In any case, you can forget about "This is basically what QFT is about" in my answer which probably elicited your comment. What I said applies to statistical field theory and that's enough physical motivation for me. $\endgroup$ – Abdelmalek Abdesselam Apr 7 '18 at 17:30