Abstract: Let $R$ be a commutative ring with 1. A row vector $u= (a_1,\cdots,a_n)\in R^n$ is called unimodular if there is another row vector $v = (b_1,\cdots,b_n)\in R^n$ such that $uv^T=1$. In other words, $a_1b_1+\cdots a_nb_n=1$. Take any matrix $M \in SL_n(R)$. The first row of $M$ is a unimodular row. Now let us ask this intriguing question: Let $u$ be a unimodular row. Is it the first row of a matrix in $ SL_n(R)$?
We shall explore this question in detail, and if possible, will talk about some recent research as well.
Video: (Not Recorded)
Date and Time: Monday, 25th September 2023, 5:30 PM - 7:00 PM (IST)
Abstract: In this lecture, a probabilistic proof of Weierstrass approximation theorem will be presented. Basic probability theory needed for the proof will be developed in a self-contained manner. The statement and motivation of the approximation theorem will also be discussed from the viewpoint of real analysis. This lecture will explain, among other things, how inter-dependent the two subjects (probability theory and real analysis) are. Special care will be taken so that this lecture is accessible to everyone.