Lattice Theory of Noncrossing Partitions and Noncrossing Arc Diagrams - Lecture 1

Explore the foundational concepts of finite lattices and their connections to Coxeter-Catalan combinatorics in this 46-minute mathematics lecture from the ESI Workshop on "Recent Perspectives on Non-crossing Partitions through Algebra, Combinatorics, and Probability." Begin with essential definitions, theorems, and classical constructions that form the basis for understanding lattice theory, even without prior knowledge of lattices and their properties. Delve into the relationship between noncrossing partitions and c-Cambrian lattices, while examining their applications in representation theory and dynamics. Designed as a self-contained presentation, gain insights into posets and prepare for advanced discussions on weak order, c-Cambrian lattices, and non-crossing partitions in subsequent lectures. Conclude with contemporary research findings and unresolved questions in this specialized field of mathematics.