Title: From Countings to Algebras: A Tale of Calabi-Yau Geometries

Abstract: In physics computing something is not necessarily the same as understanding it. The comment may well apply to the counting problem of supersymmetric gauge theories and string theory, where we encounter interesting enumerative invariants of Calabi-Yau geometries. While the countings themselves have long been discussed in terms of statistical mechanical models of crystal melting, in recent years we have learned that there are infinite-dimensional algebra underlying this counting, and this has led to fascinating interplays between physics and mathematics. In this lecture I will give an overview of this topic, and highlight some recent results and possible general lessons therein.