The arc space of a variety is introduced by John F. Nash in 1969. Nash also posed  an interesting problem so called "Nash Problem" which took 40 years to be solved. After 2000 it is unveiled that the arc space plays important roles in singularity theory and also in birational geometry. In the course I will introduce the notion of arc spaces, Nash Problem, and various application of the arc spaces into singularity theory and birational geometry.

Chapter 1 and 2 of "Algebraic Geometry" by R. Hartshorne, Springer-Verlag.

I will tell references to the students in the course.