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Invariant theory is the great romantic story of mathematics. For one hundred and fifty years, from its beginnings with Boole to the time, around the middle of this century, when it branched off into several independent disciplines, mathematicians of all countries were brought together by their common faith in invariants: in England, Cayley, MacMahon, Sylvester and Salmon, and later, Alfred Young, Aitken, Littlewood and Turnbull. In Germany, Clebsch, Gordan, Grassmann, Sophus Lie, Study; in France, Hermite, Jordan and Laguerre; in Italy, Capelli, d’Ovidio, Brioschi, Trudi and Corrado Segre; in America, Glenn, Dickson, Carus (of the Carus Monographs), Eric Temple Bell and later Hermann Weyl. Seldom in history has an international community of scholars felt so united by a common scientific ideal for so long a stretch of time. In our century, Lie theory and algebraic geometry, differential algebra and algebraic combinatorics are offsprings of invariant theory. No other mathematical theory, with the exception of the theory of functions of a complex variable, has had as deep and lasting an influence on the development of mathematics.

Let A = a < b < … be an infinite alphabet of negative letters, and let P = u 1 < u 2 < ⋯ < u n be an alphabet containing n negative places. The elements of the letterplace alphabet

I am very happy to be here before you as the Fubini Lecturer for this year, and I feel deeply honored to be given this great opportunity to share with you some of the mathematics we love.

Let A and P denote negative alphabets so that the supersymmetric algebra Super[A|P] is an ordinary polynomial algebra in the neutral elements [a|u] for a in A and u in P.

Publisher Summary This chapter describes restricted bases for finite fields. Some theorems and problems with its solution are discussed in the chapter.