Andrei Okounkov

Columbia University

March 15, 2016

These lectures will be about enumerative K-theory of curves (and more general 1-dimensional sheaves) in algebraic threefolds. In the first lecture, we will set up the enumerative problem and survey what we know and what we conjecture about it. In particular, we will meet the fundamental building blocks of the theory---threefolds fibered in ADE surfaces. In the second lecture, we will learn what geometric representation theory says about these building blocks, and, in particular, meet the present day incarnation of the Weyl group, which is really a fundamental groupoid of a certain periodic hyperplane arrangement, associated to a certain geometrically defined infinite-dimensional Lie algebra. This Weyl group completely determines the curve counts, and so seems like a very fitting topic for Hermann Weyl lectures. In the third lecture, I plan to introduce some of the geometric ideas that go into the actual technical construction of the theory.