 ## Paul Roberts

One of the classical topics in Commutative Algebra is the study of Cohen-Macaulay rings and algebras. In addition to their intrinsic interest, they have important applications to other questions, such as the Homological Conjectures. A few years ago Heitmann showed that a weaker condition, which we call the property of being almost Cohen-Macaulay, has many of the same implications and is much more likely to hold. In this talk I will define almost Cohen-Macaulay rings, give several examples, and discuss some of the implications of their existence.

 Attachments 2008-04-01roberts.pdf, courtesy of Bjorn Poonen Note: If you right click on the link to "Save file as..." and the filename shows ShowFile.htm, please rename this to the filename that you clicked on.

## Steven Sam

Let P be a lattice d-dimensional polytope. A theorem of E. Ehrhart states that the number of lattice points of nP (the nth dilate of P), as a function of positive integers n, agrees with a polynomial of degree d (the Ehrhart polynomial of P). In this talk, I will give a review of Ehrhart polynomials and discuss a paper of C. Haase and J. Schicho that classifies all such polynomials for d=2. If we replace ``lattice polytope'' by ``rational polytope,'' the word ``polynomial'' becomes ``polynomial with periodic coefficients'' (quasi-polynomials). I will discuss known bounds for the periods of Ehrhart quasi-polynomials, but show that in some cases, they are ordinary polynomials. I will also show that the classification for d=2 is incomplete if we consider rational polygons whose Ehrhart quasi-polynomial is a polynomial, and present some conjectures that I like about why some polytopes have smaller period than the known bounds suggest.

 Attachments 2008-04-01sam.pdf, courtesy of Bjorn Poonen Note: If you right click on the link to "Save file as..." and the filename shows ShowFile.htm, please rename this to the filename that you clicked on.