UC Berkeley

Commutative Algebra and Algebraic Geometry Seminar

April 01, 2008

939 Evans Hall


3:45PM: Almost Cohen-Macaulay Algebras

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.

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5:00PM: Counting lattice points and the number 2i+7

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

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