In Einstein‘s theory of general relativity, the shape of spacetime itself plays a critical role in our understanding of gravitation. Mathematically, spacetime is described by three spatial coordinates that describe position and one coordinate for time; these coordinates smoothly vary as we move from one point to another, what mathematicians call a 4-dimensional manifold, or 4-manifold. Low-dimensional topology is the study of 3 and 4-manifolds, aiming to understand the structure that remains as the manifolds are stretched or deformed without creating tears, kinks, or holes.

The study of low-dimensional topology has led to tremendous activity and cross-fertilization of ideas between mathematics and theoretical physics. One of the most important tools comes from the field of gauge theory, a technique that has proven remarkably powerful in theoretical physics in describing fundamental forces of nature. In the 1980s, it became clear that gauge theory also provided a tool to study low-dimensional topology, providing ways of distinguishing different 4-manifolds, or what mathematicians would call 4-manifold invariants. These tools have since been highly refined and harvested to great effect, leading to tremendous developments in mathematics.