Positivity, harmonicity, and random walks

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My proposal concerns three interconnected topics: 1) matrix-valued randomwalks, ergodic transformation, & W*-factors, 2) positive polynomials,power series, & polyhedra, and 3) asymptotics of coefficients of powerseries.Impact of my work:(1) While the general classification theorems for factors relating torandom walks have been around for decades, there are very few results ortechniques for deciding isomorphism or non-isomorphism of specificexamples. Results in this direction will be very significant,particularly in understanding the random walks via the dimension groupapproach.(2) Connections between problems in positivity and in real algebraicgeometry are only in their nascent stage. However, my approach seems to be very promising.(3) The results in the asymptotics section speak for themselves.Each of these topics will have plenty of questions that will be suitableas Ph.D. projects, and some as master's level projects. In working inany of the parts of the proposal, students will be exposed to a widespectrum of mathematics, much more than the usual narrow focus seen indoctoral theses.