Connectivity and percolation are two well studied phenomena in random graphs. In this talk we will discuss higher-dimensional analogues of connectivity and percolation that occur in random simplicial complexes. Simplicial complexes are a natural generalization of graphs, that consist of vertices, edges, triangles, tetrahedra, and higher-dimensional simplexes. We will mainly focus on random geometric complexes. These complexes are generated by taking the vertices to be a random point process, and adding simplexes according to their geometric configuration. Our generalized notions of connectivity and percolation use the language of homology–an algebraic-topological structure representing cycles of different dimensions. In this talk we will discuss recent results analyzing phase transitions related to these topological phenomena
