This week there’s a mini-workshop here at Columbia organized by the probabilists, on Recent Developments in Constructive Field Theory. I’ll be attending some of the talks, will write more here if I can come up with something constructive to say.
Update: One of the talks today was Sourav Chatterjee on Yang-Mills for Probabilists, explaining what Yang-Mills and lattice gauge theory are, along the lines of this preprint. The discussion brought back memories of my grad school days and early research, but I fear didn’t give much cause for optimism that there will be much progress anytime soon on a rigorous understanding of 4d Yang-Mills theory.
Update: Today Scott Sheffield gave a talk on Gauge theory and the three barriers. He started with the initial motivation of finding a continuum theory of random surfaces giving an alternative representation of gauge theory in 4d (this goes back to Polyakov and others around 1980). There’s not much progress on this, but the question has inspired a wealth of results involving in some sense random surfaces. For some details of all of this, one place to look is the website of Sheffield’s recent graduate course.