**Abstract:**

The question of finding or even effectively bounding the betti numbers of an ideal in a commutative ring is a difficult one. Even more complicated is using the structure of an ideal $I$ to find information about its powers $I^r$.

Taylor's thesis described a free resolution of any ideal generated by $q$ monomials as the simplicial chain complex of a simplex with $q$ vertices. Taylor's resolution, though often far from minimal, works for every monomial ideal $I$, giving upper bounds for the betti numbers of $\beta_{i}(I) \leq {q \choose i }$ where $q \choose i$ is the number of $i$-faces of a $q$-simplex.

If $I$ is generated by $q$ monomials and $r$ is a positive integer, then $I^r$ can be generated by $q^r$ monomials, and therefore its betti numbers are bounded by $q^r \choose i$, a number that grows exponentially.

The question that we address in this talk is: can we find a subcomplex of the $q^r$-simplex whose simplicial chain complex is a free resolution of $I^r$ for any given monomial ideal $I$ generated by $q$ monomials?

We will explore this question by considering "redundant" faces of the Taylor complex of $I^r$, which will lead us to a (much smaller) subcomplex of the Taylor complex to resolve $I^r$.

This talk is based on joint work with Susan Cooper, Sabine El Khoury, Sarah Mayes-Tang, Liana M., and Sandra Spiroff.