Constructive quantum field theory I am interested in mathematical physics concerning the formalization of quantum field theories.
Is constructive QFT a possible solution to the mass gap problem in Yang-Mills theory? If so, what is the current state of the research?
 A: The 4d Yang-Mills mass gap problem can be thought of as a problem in constructive QFT, but it does not belong exclusively to it since there are lots of different approaches for making sense mathematically and rigorously of QFT. However, this is a major open problem. By analogy with say number theory, this would be similar to big open problems like the Riemann Hypothesis, the Birch and Swinnerton-Dyer Conjecture or the global Langlands correspondence for number fields. So by definition, nobody can affirm that constructive QFT (or any other approach for that matter) is "a possible solution" for this problem. However, one should mention that there are notable partial results on the mass gap problem produced by constructive QFT. The work of Balaban as well as Magnen-Rivasseau-Sénéor gives a construction of the model without UV cutoff but with an infrared cutoff, e.g., by working on a small 4d torus. The work of Osterwalder and Seiler proves mass gap and confinement but they require a UV cutoff, e.g., their result is on the unit lattice. What is missing is a combination of both results into one where there are not cutoffs, neither UV nor IR. 
I should add that the mentioned results in the UV regime are not entirely satisfactory. The article by Magnen et al. is more of an overview than a completely detailed proof. The work of Balaban proves a stability result which is an important step but does not quite make it to the construction of correlation functions. Moreover, it spans a series of articles which totals about 1000 pages in CMP. Moving forward, it is important to simplify and improve such constructions. I think the most pressing problem for constructive QFT at present is not a frontal attack of YM4 but rather to improve renormalization group methods. In my opinion, the main problem for constructive QFT today is to develop rigorous RG methods which can handle space-dependent couplings.
