Fri, July 26, 2019
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8:00am [4:00pm] K.B. Athreya:Iowa State University Description: Probability and Statistics Seminar. Speaker: K.B. Athreya. Affiliation: Iowa State University. Date and Time: Friday 26 July, 4:00 pm - 5:00 pm. Venue: Ramanujan Hall, Department of Mathematics. Title: What can you do with one uniform random variable? Abstract: Given one uniform(0,1) random variable we show that one can generate a sequence of iid uniform r.v. and give some applications. [4:00pm] Sampat Kumar Sharma :TIFR Mumbai Description: Title: On a question of Suslin about completion of unimodular rows Abstract: R.G. Swan and J. Towber showed that if (a2, b, c) is a unimodular row over any commutative ring R then it can be completed to an invertible matrix over R. This was strikingly generalised by A.A. Suslin who showed that if (a r! 0 , a1, . . . , ar) is a unimodular row over R then it can be com- pleted to an invertible matrix. As a consequence A.A. Suslin proceeds to conclude that if 1 r! ∈ R, then a unimodular row v(X) ∈ Umr+1(R[X]) of degree one, with v(0) = (1, 0, . . . , 0), is completable to an invertible matrix. Then he asked (Sr(R)): Let R be a local ring such that r! ∈ GL1(R), and let p = (f0(X), . . . , fr(X)) ∈ Umr+1(R[X]) with p(0) = e1(= (1, 0, . . . , 0)). Is it possible to embed the row p in an invertible matrix? Due to Suslin, one knows answer to this question when r = d + 1, without the assumption r! ∈ GL1(R). In 1988, Ravi Rao answered this question in the case when r = d. In this talk we will discuss about the Suslin’s question Sr(R) when r = d − 1. We will also discuss about two important ingredients; “homotopy and commutativity principle” and “absence of torsion in Umd+1(R[X]) Ed+1(R[X]) ”, to answer Suslin’s question in the case when r = d − 1, where d is the dimension of the ring.