Hermann Weyl Lectures

Weyl groups, and their generalizations, in enumerative geometry III

Andrei Okounkov
Columbia University
March 17, 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.

Weyl groups, and their generalizations in, enumerative geometry II

Andrei Okounkov
Columbia University
March 16, 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.

Sparsification of graphs and matrices

Daniel Spielman
Yale University
November 3, 2014
Random graphs and expander graphs can be viewed as sparse approximations of complete graphs, with Ramanujan expanders providing the best possible approximations. We formalize this notion of approximation and ask how well an arbitrary graph can be approximated by a sparse graph. We prove that every graph can be approximated by a sparse graph almost as well as the complete graphs are approximated by the Ramanujan expanders: our approximations employ at most twice as many edges to achieve the same approximation factor.