As first observed by Mazur and Mumford, there existed a deep analogy between 3-manifolds and number fields. In the 90s, Morishita expanded upon and delved deeper into these connections between low-dimensional topology and number fields. More recently, Minhyong Kim proposed that, in line with the aforementioned analogy, one should develop “arithmetic field theories” for number fields. Kim’s proposal invigorated interest in this area of study, leading to a plethora of new research directions. Examples included the work of Kim and collaborators, as well as connections to the Langlands program explored by Ben-Zvi—Sakellaridis—Venkatesh. The once-unexpected comparison between 3-manifolds and number fields was now proving fruitful, and it promised to illuminate number theory, potentially leading to significant breakthroughs.

This workshop brought together experts and young researchers to further explore a field that is still largely unexplored. While the analogies were striking, we believed that the field had not yet seen its full potential. One of the goals of the workshop was to find new problems which can be attacked with ideas stemming from the analogy between number fields and 3-manifolds.

We were grateful for the financial support from the Glasgow Mathematical Journal Learning and Research Support Fund, Foundation Compositio Mathematica, London Mathematical Society, UKRI, Heilbronn Institute for Mathematical Research and Edinburgh Mathematical Society.