Invariant Theory

Invariant Theory emerged more than 170 years ago as a study of polynomials which are transformed in a prescribed way under nondegenerate linear transformations of variables. This theory went through several periods of rises and falls and nowadays it is flourishing again mainly because of deep, mutually fruitful connections with a number of disciplines (algebraic groups, Lie groups, algebraic geometry, representation theory, commutative algebra, homological algebra, Galois theory, ring theory, combinatorics, coding theory) and famous mathematical problems (14th and 13th Hilbert problems). Actually, Invariant Theory gave birth to some of them (commutative algebra and homological algebra). In the talk will be given an idea of the basic development trends and results of Invariant Theory from the beginning to the our days.