Extremal combinatorics investigates finite objects such as graphs or set systems with extremal properties. Finite geometry investigates finite incidence structures. For decades there have been interesting interactions between both areas: (1) Finite geometry provides examples of graphs and hypergraphs with extremal properties for extremal combinatorics. (2) Problems in extremal combinatorics on families of finite sets generalize naturally to questions on families of subspaces in finite vector spaces. This proposal will investigate some of these connections: (1) Low degree Boolean functions on vector spaces. (2) The investigation of q-analogs of hypergraph Turán problems. (3) Pseudorandom clique-free graphs and Ramsey numbers. These particular topics have broad relevance with applications in combinatorics and computer science.