Rational and Unirational Varieties

In this talk I will introduce the notions of rationality and unirationality from both the algebraic and the geometric point of view. Algebraically, a rational variety is one whose function field is purely transcendental, while a unirational variety admits a dominant ra- tional parametrization by projective space. Over the complex numbers, these notions also carry geometric meaning: in low dimension they are reflected in the topology and differ- ential geometry of the variety, as in the case of genus-zero curves, which are modeled by the sphere and are therefore rational. I will explain the implications between rationality, stable rationality, and unirationality, and show how these concepts fit naturally into the broader framework of birational geometry and the study of algebraic varieties.