Given a collection of sets, we will investigate what can be said about how these sets intersect one another in the presence or absence of additional structure. In the case of no further structure this is a fundamental combinatorial problem with various incarnations, such as graph and hypergraph colorings and combinatorial design theory. In the geometric setting we will encounter the main results of convex geometry, while in the topological setting such questions concern the embeddability of one space into another. We will see that the combinatorial, geometric, and topological settings should not be seen as disjoint, but that they inform one another. This viewpoint will provide us with methods to solve problems of this form and unifying results that explain recurring phenomena.

Given a collection of sets, we will investigate what can be said about how these sets intersect one another in the presence or absence of additional structure. In the case of no further structure this is a fundamental combinatorial problem with various incarnations, such as graph and hypergraph colorings and combinatorial design theory. In the geometric setting we will encounter the main results of convex geometry, while in the topological setting such questions concern the embeddability of one space into another. We will see that the combinatorial, geometric, and topological settings should not be seen as disjoint, but that they inform one another. This viewpoint will provide us with methods to solve problems of this form and unifying results that explain recurring phenomena.

Given a collection of sets, we will investigate what can be said about how these sets intersect one another in the presence or absence of additional structure. In the case of no further structure this is a fundamental combinatorial problem with various incarnations, such as graph and hypergraph colorings and combinatorial design theory. In the geometric setting we will encounter the main results of convex geometry, while in the topological setting such questions concern the embeddability of one space into another. We will see that the combinatorial, geometric, and topological settings should not be seen as disjoint, but that they inform one another. This viewpoint will provide us with methods to solve problems of this form and unifying results that explain recurring phenomena.

Given a collection of finite sets, Kneser-type problems aim to partition the collection into parts with well-understood intersection pattern, such as in each part any two sets intersect. Since Lovász' solution of Kneser's conjecture, concerning intersections of all k-subsets of an n-set, topological methods have been a central tool in understanding intersection patterns of finite sets. We will develop a method that in addition to using topological machinery takes the topology of the collection of finite sets into account via a translation to a problem in Euclidean geometry. This leads to simple proofs of old and new results.