On cut-and-join operators, matrix models and tau-functions

In my talk, I will discuss several interconnected topics related to integrable hierarchies, matrix models, enumerative geometry, and cut-and-join (or W-) operators. In particular, I will describe several examples of the exponential cut-and-join operator descriptions of the enumerative geometry generating functions, e,g., Kontsevich-Witten and Brézin-Gross-Witten tau-functions, Hermitean matrix model, and Hurwitz numbers. We will discuss their properties and comment on the relation to integrable hierarchies and matrix models. A general construction of cubic cut-and-join operators for the semi-simple cohomological field theories and topological recursion will be discussed.