Research topics in distribution theory

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 non-convex 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 one-to-one. 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 non-linear elasticity. For basic results, see [7].
• Non-linear elasticity. An approach to non-linear 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.

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, non-commutative 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 non-compact 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.

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

https://arxiv.org/abs/1511.03180