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Thursday, March 14, 2019

**Abstract:** Title: Splitting property of projective modules, by Homotopy obstructions Speaker: Satya Mandal, U. of Kansas \noindent{\bf Abstract:} Follow the link: http://mandal.faculty.ku.edu/talks/abstractIllinoisMarch19.pdf Alternate version: The theory of vector bundles on compact hausdorff spaces $X$, guided the research on projective modules over noetherian commutative rings $A$. There has been a steady stream of results on projective modules over $A$, that were formulated by imitating existing results on vector bundles on $X$. The first part of this talk would be a review of this aspects of results on projective modules, leading up to some results on splitting projective $A$-modules $P$, as direct sum $P\cong Q\oplus A$. % Our main interest in this talk is to define an obstruction class $\varepsilon(P)$ in a suitable obstruction set (preferably a group), to be denoted by $\pi_0\left({\mathcal LO}(P) \right)$. Under suitable smoothness and other conditions, we prove that $$ \varepsilon(P)\quad {\rm is~trivial~if~and~only~if}~ P\cong Q\oplus A $$ Under similar conditions, we prove $\pi_0\left({\mathcal LO}(P) \right)$ has an additive structure, which is associative, commutative and has n unit (a "monoid").

Thursday, June 13, 2019

Friday, November 1, 2019