UC Berkeley

Commutative Algebra and Algebraic Geometry Seminar

April 26, 2011

939 Evans Hall

3:45PM: Hopf Algebras and Shuffling

Persi Diaconis

Natural Hopf algebras (e.g., the free-associative algebra) give rise to natural random walks (e.g., shuffling cards). Then, an argument of Drinfeld leads to a useful diagonalization of the relevant operators. This is joint work with Amy Pang and Arun Ram.

5:00PM: What is a Matrix Factorization of a Regular Sequence?

David Eisenbud

A matrix factorization of a polynomial (or an element of a commutative ring) f is a pair of square matrices A,B with polynomial entries such that AB = BA = f*Identity. It is known that matrix factorizations exist for any polynomial, though the size of the matrices may need to be large. Matrix factorizations occur in representation theory, the study of free resolutions, and, most recently, string theory. I (and others) have been frustrated for a long time in trying to extend this notion to a situation where the single polynomial f is replaced by a nice (= regular) sequence of polynomials. I will try to explain what would constitute a ``good'' extension from my point of view, and discuss recent progress on this in joint work with Irena Peeva.

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