Abstract:
Let \(I\) in \(R = k[x_1,\dots,x_n]\) be an ideal. The Poincare-Betti series of quotient ring \(S=R/I\)
is a formal power series over a variable \(t\) with coefficients given by the dimensions of \(
\mathrm{Tor}_i(k,k) \)
over \(S\). Finding a complete characterization of what ideals \(I\) give rise to rational power series is
still an open question, although certain classes of ideals (monomial, toric, and complete intersections)
have been studied extensively. Similarly, giving formulas for minimal free resolutions of \(k\) over
\(S\) remains open in most cases. In this talk, we examine several open problems in constructing such
resolutions, and look at combinatorial approaches in the two and three variable cases. We also connect
a classification of strongly generic Artinian monomial ideals with upper intervals in the weak Bruhat
order on permutations.

This colloquium will be held in
7800 York Road,
Room 320 at 1:30 p.m., with light refreshments served after the talk, at
2:30 p.m.