## David Eisenbud

Suppose I\subset S = k[x_0,\cdots,x_n] is a homogeneous ideal. A surprising theorem of Cutkosky-Herzog-Trung, Kodiyalam, and Trung-Wang asserts that for t >> 0 the Castelnuovo-Mumford regularity of I^t is a linear function of t, say dt+e. The invariant d is relatively easy to identify, and in recent work Harris and I showed that, in a leading special case, the invariant e is connected with the regularities of fibers of a related morphism of varieties. That left---in every case---the question, ``How large does t have to be?'' I'll explain the background, and discuss a recent result from joint work with Bernd Ulrich that gives a reasonably sharp bound in the special case I treated with Harris. The work leaves open some fundamental questions about Rees algebras, and I'll discuss these as well.

## Morgan Brown

This is the first talk in RAGS. See http://math.berkeley.edu/~derman/RAG.html for a complete listing.