Mon, October 21, 2019
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- Time:
- 11:30am-12:30pm
- Location:
- Ramanujan Hall, Department of Mathematics
- Description:
- Commutative Algebra seminar I.
Speaker: Soumi Tikader.
Affiliation: ISI Kolkata.
Date and Time: Monday 21 October, 11:30 am - 12:30 pm.
Venue: Ramanujan Hall, Department of Mathematics.
Title: Orbit spaces of unimodular rows over smooth real affine algebras.
Abstract: In this talk we will discuss about the group structure on orbit
spaces of unimodular rows over smooth real affine algebras. With a few
definition and some results to start, we will prove a structure theorem of
elementary orbit spaces of unimodular rows over aforementioned ring with
the help of similar kind results on Euler class group. As a consequences,
we will prove that :
Let $X=Spec(R)$ be a smooth real affine variety of even dimension $d > 1$,
whose real points $X(R)$ constitute an orientable manifold. Then the set
of isomorphism classes of (oriented) stably free $R$ of rank $d > 1$ is a
free abelian group of rank equal to the number of compact connected
components of $X(R)$.
In contrast, if $d > 2$ is odd, then the set of isomorphism classes of
stably free $R$-modules of rank $d$ is a $Z/2Z$-vector space (possibly
trivial). We will end this talk by giving a structure theorem of Mennicke
symbols.
- Time:
- 3:00pm-4:00pm
- Location:
- Room No 216 Department of Mathematics
- Description:
- Combinatorics seminar.
Speaker: Projesh Nath Choudhury.
Affiliation: IISc Bengaluru.
Date and Time: Monday 21 October, 3:00 pm - 4:00 pm.
Venue: Room 216, Department of Mathematics.
Title: Distance matrices of trees: invariants, old and new.
Abstract: In 1971, Graham and Pollak showed that if $D_T$ is the distance
matrix of a tree $T$ on $n$ nodes, then $\det(D_T)$ depends only on $n$,
not $T$. This independence from the tree structure has been verified for
many different variants of weighted bi-directed trees. In my talk:
1. I will present a general setting which strictly subsumes every known
variant, and where we show that $\det(D_T)$ - as well as another graph
invariant, the cofactor-sum - depends only on the edge-data, not the
tree-structure.
2. More generally - even in the original unweighted setting - we
strengthen the state-of-the-art, by computing the minors of $D_T$ where
one removes rows and columns indexed by equal-sized sets of pendant nodes.
(In fact, we go beyond pendant nodes.)
3. We explain why our result is the "most general possible", in that
allowing greater freedom in the parameters leads to dependence on the
tree-structure.
4. Our results hold over an arbitrary unital commutative ring. This uses
Zariski density, which seems to be new in the field, yet is richly
rewarding.
We then discuss related results for arbitrary strongly connected graphs,
including a third, novel invariant. If time permits, a formula for
$D_T^{-1}$ will be presented for trees $T$, whose special case answers an
open problem of Bapat-Lal-Pati (Linear Alg. Appl. 2006), and which extends
to our general setting a result of Graham-Lovasz (Advances in Math. 1978).
(Joint with Apoorva Khare)