Commutative Algebra Seminar

University of Utah, Department of Mathematics




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Date: Friday 3/23/12, 1:00-2:00pm in LCB 215

Courtney Gibbons (Univ. Nebraska-Lincoln),

Title: Rational multiples of Betti diagrams
Abstract: In 2006, Boij and Soederberg offered a new and exciting way to study modules over a polynomial ring by looking at their Betti diagrams up to nonnegative rational multiple. They then used convex geometry to gain new insights. I'll talk about joint work with Christine Berkesch, Jesse Burke, and Daniel Erman where we use some of these ideas to study the Betti diagrams of modules over a one-dimensional quadric hypersurface.

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Date: Friday 3/23/12, 2:00-3:00pm in LCB 215

Branden Stone (Univ. of Kansas),

Title: Countable CM type and Isolated Singularities
Abstract: Witheld for suspense.

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Date: Friday 4/13/12, 12:00-1:00pm in JWB 333

Jack Jeffries (Univ. of Utah),

Title: Finite F-Representation Type and F-Signature
Abstract: In 1999, an investigation of differential operators in positive characteristic was led by Smith and Van den Bergh to define the notion of rings of finite F-repretation type. This property has many interesting connections with current research topics, but many open questions remain. We discuss some of the consequences of F-finite representation type, including a new result on the F-signature of such rings.

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Date: Friday 4/13/12, 1:00-2:00pm in JWB 333

Lance Miller (Univ. of Utah),

Title: Hilbert-Kunz functions of rank 1 matrices.
Abstract: E. Kunz in 1976 introduced a positive characteristic variant of Hilbert-Samuel theory where powers of an ideal are replaced by Frobenius powers. Unfortunately, he erroniously conclucded the corresponding limit defining the multiplicity does not exist! P. Monsky later showed the error in Kunz's conclusion and how the associated multiplicity, called the Hilbert-Kunz multiplicity, is deeply related to singularity theory and tight closure. This multiplicity is the first coefficient of a function called the Hilbert-Kunz function, which is not in general polynomial like. This talk will introduce Hilbert-Kunz theory and discuss some work on computing the Hilbert-Kunz functions of determinantal varieties via Grobner basis. This is joint work with I. Swanson.

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Fall 2011 Spring 2011 Fall 2010