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Milestone Year
1983
Real semi-algebraic geometry in Dynamics, Analysis, and Number Theory
In 1983 a new approach has been developed to a range of problems in Analysis and Dynamics, which succeeded to combine the power of Real Semi-Algebraic Geometry with Approximation Theory. The first achievement of this appraoch was a "quantitative" version of the classical Sard Theorem, and a number of related results. The most important, up to day, result was obtained in 1986, and consists of a solution of an outstanding open problem in smooth Dynamics, known as the Entropy conjecture. This result remains a major tool in Dynamics up to the current day.
In 1991 this result has been extended to Analytic Dynamics. Since 1986 some of the tools and methods developed between 1983-1991 have found new (and sometimes pretty unexpected) applications in Analysis, Dynamics, Geometry, and Geometric Computing. Recently a new exciting application of one of the main such tools, the so-called "Gromov-Yomdin Algebraic Lemma", has been used in Model Theory (a branch of Mathematical Logic). This development ultimately lead to the creation of a new powerful tool in Geometric Number Theory, where solutions of several longstanding open problems have been found.