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Invited Paper Session Abstracts - Commutative Algebra

Saturday, August 3, 9:00 a..m. - 11:50 a.m., Duke Energy Convention Center, Room 200

Commutative algebra is central discipline at the intersection of algebraic geometry, number theory, and many other areas. Many of the foundations were laid by Emmy Noether and modern commutative algebra combines techniques from computational symbolic algebra, combinatorics, graph theory, and homological and homotopical algebra. This session will have expository talks on many flavors of commutative algebra with a broad appeal towards the subjects natural influence. The talks will be aimed at a general mathematical audience, will be suitable for both students and faculty, and will hope to expose participants to the rich tapestry of current and classic results available.

Irena Swanson, Reed College
Lance Miller, University of Arkansas, Fayetteville

Convergence of Rees Valuations

9:00 a.m. - 9:20 a.m.
Matthew Toeniskoetter, Florida Atlantic University


Associated to every ideal in a Noetherian ring, there is a finite set of divisorial valuations called Rees valuations which arise naturally through the integral closure of powers of that ideal and through the blow-up of that ideal. For a Noetherian local domain, we repeatedly blow up the maximal ideal to obtain an infinite sequence of Noetherian local domains, and if the maximal ideal of each of these rings has a single Rees valuation, we show that the corresponding sequence of Rees valuations converges. We demonstrate examples with polynomial rings and give explicit descriptions to the valuations we get in this case.


An Algebraic Condition that Allows Us to Do Intersection Theory

9:30 a.m. - 9:50 a.m.
Patricia Klein, University of Kentucky


A classical theorem in intersection states that if X and Y are two curves in three variables defined by homogeneous polynomials whose greatest common divisor is a constant, then the number of points of intersection of X and Y is equal to the product of the degrees of the polynomials that define them. This result is known as Bézout's theorem and was first published by Bézout in 1779. If one wants to generalize Bézout's theorem to higher dimensions, it is not obvious what the right analogue of the greatest common divisor being a constant should be. We will discuss some examples to explore possibly appropriate conditions and end by describing the condition known as Cohen-Macaulay, which is of wide interest not only to those working in intersection theory but to those working in virtually all areas of commutative algebra.


On Flavors of Factorization in Commutative Rings with Zero Divisors

10:00 a.m. - 10:20 a.m.
Ranthony Edmonds, Ohio State University


Factorization theory is concerned with the decomposition of mathematical objects. One of the earliest and most significant results involving factorization is the Fundamental Theorem of Arithmetic, which states that every integer can be written uniquely as the product of primes. Thus we can think of prime numbers as the atoms of the integers. We can generalize this idea of unique factorization into atoms to a commutative ring called a unique factorization domain.

Many early questions related to factorization in commutative rings were concerned with how close a ring is to having unique factorization. In 1990 Anderson et all presented a general theory of factorization properties in commutative rings weaker than unique factorization, which was then extended some years later to commutative rings with zero divisors. In this talk we discuss some issues that arise when working with zero divisors, and the different flavors of factorization in commutative rings with zero divisors that emerged to remedy these issues. Some techniques for constructing counterexamples in rings with zero divisors will also be described.


Direct-sum Decompositions of Modules: The Good, the Bad, and the Ugly (aka Interesting)

10:30 a.m. - 10:50 a.m.
Nicholas Baeth, Franklin and Marshall College


In most scientific endeavors, one wants to classify the basic building blocks of objects and then describe how all objects can be constructed in terms of these atoms. In linear algebra we learn that every \(n\)-dimensional vector space over a field \(F\) is really just \(F^n\) --- a direct sum of \(n\) copies of the field \(F\). It is then possible to study any finite-dimensional vector space using this nice identification. Modules are the generalization of vector spaces when the field \(F\) is replaced by a ring \(R\). While many nice vector space-like properties carry over to module theory, many do not. In particular, most \(R\)-modules do not decompose as a sum of copies of the ring \(R\), and even over very nice rings it can be difficult to identify the "indecomposable" modules. While over some rings, all modules decompose in a unique way as a direct sum of indecomposable modules, this is not always the case. In this talk we will give a brief history of the study of direct-sum decompositions over certain kinds of nice rings and illustrate, by way of examples, how good, bad, and ugly (or interesting, depending on your point of view) direct-sum decompositions of modules can be.


Syzygy - When Submodules Align

11:00 a.m. - 11:20 a.m.
Courtney Gibbons, Hamilton College


In astronomy, a syzygy is an alignment of celestial bodies. In mathematics, a syzygy is an alignment of a kernel of one homomorphism with the image of another! In this talk I'll introduce free resolutions, syzygies, and a few applications thereof.