Counting incompressible surfaces in 3-manifolds

A 3-manifold is small, smallish or very large if it contains no embedded incompressible surfaces (up to isotopy), finitely many, or infinitely many. We will prove a structure theorem for counting
such surfaces by Euler characteristic in a cusped hyperbolic 3-manifolds. Our proofs use PL methods (ideal triangulations, normal and almost normal surfaces), simple isotopies, hyperbolic geometry, come with a computer-implementation, illustrate phenomena for triangulated 3-manifolds with at most 18 ideal tetrahedra and connect with quantum 3-manifold invariants (the q-series of the 3D-index of Dimofte-Gaiotto-Gukov). Joint work with Nathan Dunfield and Hyam Rubinstein.