Let G be a simple algebraic group over an algebraically closed field. It was proved by the author [in Fundam. Math. 95, 173-188 (1977; Zbl 0363.02060)] that the group-language theory of G is categorical in uncountable powers. This result implies that the algebraic-geometrical structure of such groups is defined by the abstract group structure. A more exact version of the last fact is given by a theorem of A. Borel and J. Tits [Ann. Math., II. Ser. 97, 499-571 (1973; Zbl 0272.14013)]. The known proofs of this theorem use the deep structural theory of simple algebraic groups. The author offers an easier model- theoretic proof of the theorem.
One of the purposes of the paper is to show how model-theoretical methods can work in algebraic geometry. ”There are two basic facts which link the general algebraic geometry with model theory. First,...all structures definable in an algebraically closed field... are \(\omega\)-stable of finite Morley rank....The Morley rank is a good analogue of the algebraic-geometrical dimension and in most cases coincides with it. The second fact is a theorem of A. Tarski, which states that every relation definable in an algebraically closed field is a Boolean combination of polynomial equations.”