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.

Peter, what makes you say “I fear [it] didn’t give much cause for optimism…”

Isn’t translating and explaining the problems clearly to mathematicians a good thing?

Bill,

Definitely a good thing. I just meant that, from the discussion amongst experts in the field, things haven’t advanced that much since my grad student days and it seems that we’re still very far from a solid understanding of basic issues about pure Yang-Mills in 4d.

Dear Peter,

Thank you for advertising our little workshop. But, man, that was a bit harsh: “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”. If this was a workshop on number theory, it would be like saying “but I fear didn’t give much cause for optimism that there will be much progress anytime soon on a rigorous understanding of the Riemann Hypothesis”.

Anyway, I thought the meeting was quite exciting with lots of interesting new results and prospects for more in the near future.

Abdelmalek Abdesselam,

Sorry if that came across as harsh, your analogy with the Riemann hypothesis is appropriate.

In the end, what struck me from the talks that I attended was the way the Yang-Mills question has inspired a range of new mathematics, answering other questions although not the Yang-Mills one. Maybe someday this activity will develop new ideas that will ultimately provide the needed new insight into Yang-Mills.

Also worth pointing out is that to a large extent it seems that physicists have given up on the Yang-Mills problem (even on finding a useful heuristic calculational method, much less something rigorous). That mathematicians are willing to keep studying this and trying to find something new is encouraging.