Many fundamental mathematical structures, such as the rationals or the random graph, are homogeneous, meaning that local isomorphisms extend to global automorphisms. Such structures arise as limits of classes of finite structures and encode these classes in a single object. This viewpoint has proved fruitful in model theory, universal algebra, and computer science, with applications to constraint satisfaction, automata theory, and verification. Homogeneous structures have rich automorphism groups, which makes them interesting for topological dynamics. For many applications, however, automorphism groups do not store enough information about the homogeneous structure, and one must instead consider polymorphism clones. Universal algebra has recently achieved major results for polymorphism clones on finite structures, culminating in the 2017 resolution of the Feder–Vardi dichotomy conjecture. An analogous conjecture for homogeneous structures remains open despite growing structural insights.
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