1. Title: Sequences of 0's and 1's: Hahn Properties

Speaker: Dr. Maria Zeltser,

Tallin University, Estonia

Day, Date and Time: Tuesday, April 1, 2008, 4.00-5.00 p.m.

Venue: Ramanujan Hall, Dept. of Mathematics

2. Title: Gorenstein Approximation, Dual Filtrations and Applications

Speaker: Dr. Tony J. Puthenpurakal,

Department of Mathematics, IIT-Bombay

Day, Date and Time: Tuesday, April 1, 2008, 2:30-3:30p.m.

Venue: Ramanujan Hall, Dept. of Mathematics

Abstracts

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1. Abstract for Dr. Zeltser's talk

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The main idea of the talk is the following: for a given sequence

space we have a property which is satisfied for a simple and small

subset of it -- for the set of all sequences of 0's and 1's in this

space. We ask whether the whole space has this property.

For example, in 1922 Hahn proved that if an (infinite) matrix

sums all sequences of 0's and 1's, then it sums all bounded

sequences. (In summability the term

``sums''

means that the matrix transforms the given sequence to a convergent

sequence.) So if the matrix maps the set of all 0-1 sequences to the space

of all convergent sequences $c$, then it also maps the space of all

bounded sequences to $c$.

We would like to study whether this result remains true if we replace the

space of all bounded sequences by any sequence space $E$ and the set of

all 0-1 sequences by the set of all 0-1 sequences in $E$. In this case we

say that $E$ has the matrix Hahn propoerty. We consider also two

generalizations of this notion.

2. Abstract for Dr. Puthenpurakal's talk

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We give a two step method to study certain questions regarding associated

graded module of a \CM \ module \wrt \ an $\m$-primary ideal $\A$ in a

complete Noetherian local ring $(A,\m)$. The first step, we call it

Gorenstein approximation, shows that it suffices to consider the case when

both $A$, $ \GA = \bigoplus_{n \ge 0} \A^n/\A^{n+1} $ are Gorenstein and

$M$ is a maximal \CM \ $A$-module. The second step consists of analyzing

the classical filtration $\{ \Hom_\A(M,\A^n) \}_{\nZ}$ of the dual

$\Hom_A(M,A)$. We give many applications of this point of view. For

instance we show that if $(R,\n)$ is \CM \ then the a-invariant of

$G_\n(R)$ is $-\dim R$

\ff \ $R$ is regular local. We also extend to modules a result of Ooishi

relating symmetry of $h$-vectors and the Gorenstein property of associated

graded rings.

Tony J. Puthenpurakal

Convener

SCC