Perlen-Kolloquium: Ivan Cheltsov

Rational and non-rational varieties

Pythagorean triples are given by rational parametrization of the unit circle. Similarly, all conics in the plane admits a similar parametrization. This is no longer the case for the curves of higher degrees. 

For example, non-singular plane cubic curves (elliptic curves) cannot be rationally parametrized. In dimension one and two, the problem of existence of such parametrization is completely solved.

In higher dimensions, the picture is less clear. In this talk, I will survey know results and open problems in this direction.

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