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KUMUNU-03 2001-09-29

KUMUNU  3  -  September 29,  2001

The talks will be given in 120 Snow Hall.

Time Speaker Title               
11:00 - 11:40 G. Leuschke The rational signature of a ring of prime characteristic
11:50 - 12:30 N. V. Trung Cohomological degree and Castelnuovo-Mumford regularity
12:30 - 2:20 Lunch break in 406 Snow Hall
2:20 - 3:00 R. Miranda Linear Systems of Plane Curves: some combinatorial approaches
3:10 - 3:50 J. Walker A structural approach to coding theory
4:00 - 4:40 B. Harbourne Open Problems and Recent Progress on Problems related to Lagrange Interpolation
4:50 - 5:30 S. Iyengar Gorenstein objects in algebra and topology
6:30 -  Dinner party at C. Huneke's house

The rational signature of a ring of prime characteristic

Graham Leuschke

In the representation theory of local rings $R$ of characteristic $p$, a central problem is the direct-sum decomposition of the ring of $q$th roots, $R^{1/q}$, as $q$ varies over powers of $p$. The ring $R$ is called F-finite if $R^{1/q}$ is a finitely generated $R$-module. When $R$ is regular and F-finite, $R^{1/q}$ is of course a free module; the converse is a well-known theorem of Kunz. When $R$ is Cohen-Macaulay and F-finite, each direct summand of $R^{1/q}$ is a maximal Cohen-Macaulay $R$-module. This will be the situation in my talk.

K.E. Smith and M. van den Bergh consider the following property: $R$ has "finite F-representation type" (FFRT) if only finitely many isomorphism classes of indecomposable maximal Cohen-Macaulay modules $M$ appear as direct summands of $R^{1/q}$, as $q$ ranges over all powers of $p$. They show that if $R$ has FFRT and is strongly F-regular, then the number of copies of $M$ splitting out of $R^{1/q}$ divided by $q^{\dim(R)}$ has a positive limit.

I will talk about a numerical invariant of the ring $R$ defined similarly. Let $d = \dim(R)$, and let $a_q$ be the rank of a maximal free summand splitting out of $R^{1/q}$. Then the "rational signature" of $R$ is $s(R) := \lim_q a_q/q^d$, provided the limit exists. This number measures the F-rationality of $R$: when $R$ is Gorenstein, the limit always exists, and is positive precisely when $R$ is F-rational. The rational signature also carries more subtle information about the singularity associated to $R$, especially when $R$ is a ring of invariants of a finite group.

These results are from joint work with Craig Huneke

Cohomological degree and Castelnuovo-Mumford regularity

N. V. Trung 

This is a report on a joint work with M. Rossi and G. Valla which shows that the Castelnuovo-Mumford regularity of the tangent cone of a local ring $A$ is effectively bounded by the dimension and any cohomological degree of $A$. From this it follows that there are only a finite number of Hilbert-Samuel functions of local rings with given dimension and cohomological degree.

Linear Systems of Plane Curves: some combinatorial approaches

Rick Miranda

Fix general points p_1,...,p_n in the plane, fix multiplicities m_1,...,m_n, and consider the linear system L of all plane curves of degree d having multiplicity at least m_i at p_i for each i. In this generality it is not known what the dimension of the linear system L is, although there are conjectures (due to Segre, Harbourne and Hirschowitz) which are open. I will discuss some combinatorial approaches which give reductions of this problem in certain cases, and which have been successful in determining the dimension of L in some otherwise intractable situations. The approach is based on work of Lorentz and Lorentz.

A structural approach to coding theory

Judy Walker

Abstract: After a brief introduction to the theory of error-correcting codes, we will focus on a structural approach to the topic. This approach began with D. Slepian in the late 1950's (remarkably, since at least by some measure, coding theory wasn't born until 1948!). In fact, Slepian developed all the definitions one would need to consider the Grothendieck ring of the category of binary linear codes. It was Slepian's intent to show that every code is equivalent to a code which has a generator matrix in a certain canonical form. Alas, this is not true, but Slepian did manage to make some significant contributions nonetheless. Perhaps most importantly, he developed the idea of an indecomposable code, that is, a code which is not isomorphic to a nontrivial direct sum of two other codes.

The major breakthrough came in the late 1990's when E.F. Assmus introduced the notion of critical indecomposable codes. There is a quasi-canonical form for the generator matrix of a critical indecomposable code, giving a partial solution to Slepian's problem. Further, every indecomposable code maps onto a critical indecomposable code of the same dimension, and there are only finitely many critical indecomposable codes of each dimension.

Since every code is a direct sum of indecomposable codes and every indecomposable code is ``built'' from a critical indecomposable code, critical indecomposable codes are the ``building blocks'' of coding theory. By studying these objects, one can obtain considerable insights. As an example, we will discuss self-dual codes from this point of view.

Open Problems and Recent Progress on Problems related to Lagrange Interpolation

Brian Harbourne

Abstract: Lagrange Interpolation involves finding a polynomial with given values and derivatives. In the simplest situation, one wants a polynomial such that it and all of its derivatives up to order m-1 vanish at n given points. We can ask: What is the least degree a(n,m) of a nontrivial such polynomial if the n points are taken as general as possible?

This is easy to answer for polynomials of one variable, but it leads to a range of open problems for polynomials of two or more variables. In this talk, we'll look at some of these open problems and recent progress in the case of 2 variables:

What is known about the value of a(n,m)? What can be said asymptotically (i.e., lim_{m\to\infty} a(n,m)/m or lim_{n\to\infty} a(n,m)/\sqrt{n})?

Gorenstein objects in algebra and topology

Srikanth Iyengar

Abstract: One of the salient features of the cohomology of a compact manifold is that it satisfies Poincare' duality. This leads one to label any finite dimensional graded connected algebra over a field with such a duality property a Poincare' algebra. When one considers (finite dimensional graded connected) commutative algebras, the Poincare' algebras are precisely the ones that are Gorenstein. Therefore, one may consider a general (commutative) Gorenstein ring as a higher dimensional analogue of a Poincare' algebra. This in turn suggests a way of arriving at a meaningful notion of higher dimensional analogues of spaces, and also of not-necessarily-commutative algebras, with Poincare' duality. However, there are a myriad properties that characterize commutative Gorenstein rings, not all of which lend themselves to such extensions, and even those that do lead to different classes of 'Gorenstein rings'. This is perhaps a blessing for we may choose one that best suits our needs. In this talk I hope to describe some aspects of one such extension which is particularly tailored to fit rings that arise in topology and provides a natural generalization of the classical Poincare' duality for manifolds; it also raise some new issues in commutative algebra which may prove to be of independent interest. This is part of work in progress with Bill Dwyer and John Greenlees.

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