Mon, October 21, 2019
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October 2019
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11:00am [11:30am] Soumi Tikader :ISI Kolkata
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.

3:00pm [3:00pm] Projesh Nath Choudhury: IISc Bengaluru
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)