Title: Hilbert functions in algebra and geometry
Abstract: In a famous 1890 paper, David Hilbert shook the
world of algebra by proving that for any graded module over a
polynomial ring the vector space dimensions of its graded components
are eventually given by a polynomial function. The function that
records these dimensions is nowadays known as the Hilbert function
in his honor.
In this talk, we will explore the properties of Hilbert functions and
discuss some open problems concerning them that have puzzled algebraists
for a while. Many questions on Hilbert functions can be reduced to the
zero-dimensional case, that is, to investigating the Hilbert functions
of a finite set of points in projective space. This will lead us towards
some fruitful connections to geometry.