Valuations, integral dependence, and multiplicities

The theory of integral closure of ideals, originating in the early twentieth century with work of Krull, Zariski, Rees, and others, remains a vibrant area of research in commutative algebra. This theory's significance stems from its connections with valuations, which enable a wide range of applications in other mathematical fields such as combinatorics, algebraic geometry, and number theory. Asymptotic properties of integral closures can be understood through multiplicities, which have their origins in intersection theory, developed by Hilbert, Noether, Van der Waerden, and others. In the 1950s, advancements by Samuel, Serre, and Rees brought multiplicities to the forefront of commutative algebra, leading to extensive further development in various directions, including recent works. This talk will provide an overview of the history and key results in these areas, highlighting some open problems along the way.