Title: counting essential surfaces in 3-manifolds
Abstract: Counting embedded curves on a surface as a function of their length has been much studied by Mirzakhani and others. I will discuss analogous questions about counting surfaces in a 3-manifold, with the key difference that now the surfaces themselves have more intrinsic topology. As there are only finitely many essential surfaces of bounded Euler characteristic up to isotopy in an atoroidal 3-manifold, it makes sense to ask how the number of isotopy classes grows as a function of the Euler characteristic. Using Hakenâ€™s normal surface theory, we can characterize not just the rate of growth but show the exact count is a quasi-polynomial. Moreover, our method allows for explicit computations in reasonably complicated examples.
This is joint work with Stavros Garoufalidis and Hyam Rubinstein and the reference is our paper with the same title in Inventiones (2022). The only background I will assume is the notion of a manifold, the genus of a surface, and a little about the fundamental group;