Gauge Fields in Arithmetic, Topology and Physics

As first observed by Mazur and Mumford, there exists 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 include 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 is now proving fruitful, and it promises to illuminate number theory, potentially leading to significant breakthroughs. This workshop will bring together experts and young researchers to further explore a field that is still largely unexplored. While the analogies are striking, we believe that the field has not yet seen its full potential. One of the goals of the workshop is to find new problems which can be attacked with ideas stemming from the analogy between number fields and 3-manifolds.