2023 Program for Women and Mathematics: Patterns in Integers: dynamical and number theoretic approaches

Abstract: A central question in extremal graph theory is to determine ex(n,H), the maximum number of edges in a n-vertex graph containing no subgraph isomorphic to some forbidden graph H. One may define a topological variant of this extremal quantity as follows. Let S be a fixed surface (e.g. the sphere, torus, klein bottle, etc.), and let ex_hom(n,S) denote the maximum number of 2-simplices in an n-vertex simplicial complex in which no sub-complex is homeomorphic to S. I will discuss recent results that completely determine the asymptotics of ex_hom(n,S) for any fixed surface S, as well as some techniques for bounding ex_hom(n,X) for arbitrary 2-dimensional simplicial complexes X.