A trisection of a smooth 4-manifold X is a decomposition of X into three simple 4-manifolds that intersect along simple 3-manifolds. They were introduced in 2012 by Gay and Kirby in an attempt to generalize Heegaard splittings of a 3-manifold into four dimensions, and indeed many of the basic notions and results relating to Heegaard splittings have analogues in the context of trisections. In particular, one can reduce the data of a trisection to a diagram on a surface and use this to study smooth 4-manifolds combinatorially.

This talk will be expository in nature: we will spend the first part reviewing Heegaard splittings and stating the standard existence and uniqueness results. The rest of the talk will be devoted to defining trisections and trisections diagrams, stating and explaining the analogous existence and uniqueness results, and giving as many examples as possible.