In 2001, the Scuola Normale Superiore di Pisa started a tradition of regular colloquia “Colloquio De Giorgi” held once a month and providing (non-technical) overviews of certain fields of mathematics. Recently, invited speakers have prepared written versions of their talks. This booklet contains, for the first time, written expositions of talks presented in the year 2006. There is also a list of talks from previous years summarizing the history of the colloquia from its beginnings.

In the book, the reader can find seven lectures by Y. F. Bilu (on Diophantine equations with separated variables), C. De Concini (on certain branching rules for quantized enveloping algebras), S. Gindikin (on a role of the Cauchy integral formula in harmonic analysis on symmetric Stein manifolds), D. Goldfeld (on the history of the Gauss class number problem), D. Masser (on polarizations on Abelian varieties), Z. Rudnick (on the statistical distribution of eigenvalues of the Laplacian on compact manifolds) and L. Szpiro and T. J. Tucker (on dynamics of maps on an algebraic variety). The book offers a very nice collection of (short) reviews of many interesting fields in mathematics.

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