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How irrational is an irrational variety? Recall that an algebraic variety X is said to be rational if it contains a Zariski open subset isomorphic to a Zariski open subset of projective space. There has been a great deal of recent activity and progress on issues of rationality, but most varieties aren’t rational. I will survey a body of work concerned with a complementary question, namely measuring and controlling “how irrational” a non-rational variety might be. The talk will be aimed at a general mathematical audience.

Measures of association between algebraic varieties. I will discuss some joint work with Olivier Martin that attempts to measure “how far from birationally isomorphic” two varieties X and Y of the same dimension may be. The idea is to study the minimal complexity (in various senses) of correspondences between them.

Further developments and open problems. I will survey some further developments on these matters, and discuss some of the many open problems that present themselves.