#### 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.