The long-standing classification problem in the theory of Heegaard splittings of -manifolds is to exhibit for each closed -manifold a complete list, without duplication, of all its irreducible Heegaard surfaces, up to isotopy. We solve this problem for non-Haken hyperbolic -manifolds.
"On the classification of Heegaard splittings." Duke Math. J. 167 (15) 2833 - 2856, 15 October 2018. https://doi.org/10.1215/00127094-2018-0023