MSI Weekly Bulletin - Week starting Monday 7 August, 2006

Bear witness to a near murder! Weyl wrote that when Hilbert proved in 1892 that elements of polynomial rings invariant under the action of certain groups were finitely generated, he \"solves the main problems and thus almost kills the whole subject\" of classical invariant theory. In this talk, I detail a proof of this powerful result. Along the way, I detail two of the crucial tools needed, proving Hilbert\'s basis theorem, and offering some remarks on the Reynolds operator.

This year marks the 150th anniversary of the birth of Andrei Andreevich Markov (1856 --1922), and the 100th anniversary of his first paper on what came to be called Markov chains. These are probabilistic models which allow for statistical dependence between observations on the state of a system evolving over time. The underlying theory rests on properties of matrices with non-negative entries and row sums unity (stochastic matrices). Myriad applications range from sociology to physics, and recently to the search engine Google. Markov\'s motivation for his first paper on this topic was to provide a counterexample to an assertion of his old adversary, the mathematician Pavel Alekseevich Nekrasov (1853 -- 1924), which attempted to link the statistical regularity observed in human affairs with the theological concept of free will. This presentation will focus on the life and times of Markov (who achieved great eminence in number theory as well as in probability), on elements of stochastic matrix underpinnings of Markov chains, and on the Markov chain formulation of the PageRank algorithm, which underlies the Google search engine.