Expansion and Minors in Hypergraphs Uli Wagner ETH Zürich Let X be a 3-uniform hypergraph, or in other words, a 2-dimensional simplicial complex. We discuss several notions of what it means for X to be expanding, with a particular focus on a generalization of edge expansion that arose in recent work of Gromov (and, independently, of Linial and Meshulam). We also discuss a new notion of minors for hypergraphs/simplicial complexes, called homological minors. The definition is motivated by extremal problems relating the combinatorics of a hypergraph (the number of hyperedges/faces) to its topological properties such as (non)embeddability in Euclidean spaces or the existence of Tverberg-type partitions. We then relate the two notions and show that every ε-expanding 2-dimensional hypergraph with C triangles incident to every edge (on average) contains a large 2-dimensional complete minor on t vertices (provided ε and C are sufficiently large in terms of t). (While this statement is what one would expect in view of analogous statements for graphs, the arguments in higher-dimensions are rather different.) As a corollary, one also obtains asymptotically sharp bounds for the threshold probabilities at which random complexes stop being embeddable and start having Tverberg-type substructures. All of these happen at p = Θ(1/n) (with a constant depending on the precise context).