Beijing Institute of Mathematical Sciences and Applications

In this course I plan review what is known about quasiclassical tau-function, introduced by Igor Krichever in 1992 in the context of topological string theories. The first aim is to define it and formulate its main properties, based on some fundamental facts from the complex geometry of Riemann surfaces, mainly the Riemann bilinear identities. I am going to present some simple proofs, and discuss most important examples, which include the Seiberg-Witten prepotentials and matrix models. I am going to pay special attention to one of the most important points: the residue formula for the third derivatives of the Krichever tau-function and its relation with associativity or WDVV equations.
Finally I would like to discuss certain modern developments, related with this object, which include the relation with instanton partition functions, isomonodromic tau-dunctions and even some unexpected relations with other famous relations in mathematical physics.