|Location:||MSRI: Simons Auditorium|
The Hilbert function and the graded Betti numbers are invariants we use to study graded rings and modules. When we look at graded modules generated in degree zero over the standard graded polynomial ring, we know by Macaulay's classification exactly what the possible Hilbert functions are. However, for the graded Betti numbers, we are far from having such a complete description. What we do have, is a clear picture of what the set of Betti tables looks like up to scaling, i.e., the cone spanned by all Betti tables in a suitable vector space over the rational numbers. I will explain this picture and give an overview of the sequence of steps that led to it.
When changing the grading, we don't even know much about the possible Hilbert functions and it therefore interesting to study also Hilbert functions up to scaling.