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Description

A trisection is a decomposition of a four-manifold into three trivial pieces and serves as a four-dimensional analogue to a Heegaard decomposition of a three-manifold.  In this talk, I will discuss an adaptation of the theory of trisections to the relative setting of knotted surfaces in the four-sphere that serves as a four-dimensional analogue to bridge splittings of classical knots and links - every such surface admits a decomposition into three standard pieces called a bridge trisection.  I'll describe how every such decomposition can be represented diagrammatically as a triple of trivial tangles and give a calculus of moves for passing between diagrams of a fixed surface.  This is joint work with Alexander Zupan.