## Departmental Colloquium

Izzet Coskun

UIC

The Hilbert Scheme of Points

**Abstract:**The Hilbert scheme of $n$-points $X^{[n]}$ is a `compactification' of the set of unordered $n$-tuples of distinct points on a projective manifold $X$. When the dimension of $X$ is two, $X^{[n]}$ is a smooth, projective manifold with many remarkable properties. Consequently, it plays a central role in many areas of mathematics ranging from representation theory to algebraic geometry and from algebraic combinatorics to symplectic geometry. For example, $X^{[n]}$ provide examples of holomorphic symplectic manifolds when $X$ is a holomorphic, symplectic surface. The homology of $X^{[n]}$ has the structure of an irreducible representation of the Heisenberg superalgebra. Similarly, $X^{[n]}$ play a central role in resolutions of canonical surface singularities. In this talk, I will give a brief introduction to the Hilbert scheme of points and its many amazing properties.

Friday September 14, 2012 at 3:00 PM in SEO 636