What is...?

For a low-dimensional manifold, one often tries to understand its intrinsic topology and geometry through its submanifolds, in particular of co-dimension 1. To be interesting and to give some information, such a submanifold should interact with the 3-manifold in a non-trivial way. In particular, it was noticed before that presence of embedded incompressible surfaces in a 3-manifold can give information about that manifold. However to construct, classify or count such surfaces is a non-trivial task. We will start by explaining what is behind the definitions, and, if the time allows, will eventually discuss a universal upper bound for the number of non-isotopic genus g surfaces embedded in a hyperbolic 3-manifold, polynomial in hyperbolic volume of the 3-manifold. This touches on ongoing joint work with Marc Lackenby, another recent joint work with Jessica Purcell, and earlier joint work with Joel Hass and Abigail Thompson.