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
