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 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.