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Explore the fundamental concepts of congruences on finite lattices through this mathematical lecture delivered at the Centre International de Rencontres Mathématiques. Begin with essential facts about lattice congruences that every combinatorialist should understand, with particular emphasis on ideas most relevant to the weak order on finite Coxeter groups and framing lattices. Apply these theoretical foundations to the weak order through motivating examples while developing the combinatorics of congruences and quotients in both general and specific contexts. Discover how lattice theory connects to the geometric setting of hyperplane arrangements, posets of regions, and shards when time permits. This presentation was recorded during the thematic meeting "Beyond Permutahedra and Associahedra" and provides comprehensive coverage of lattice congruence theory with direct applications to combinatorial mathematics.
