The January 21, 2014 O.M.N.I. Talk: Lance Miller presents

Hilbert's 14th problem: Finite generation
of Rings of Invariants.

This talk concerns Hilbert's famous question on finite generation of rings of invariants, which appeared as the 14th problem in his famous 23 problems proposed at beginning of the 1900s. The positive answers to this question generated a substantial amount of mathematics that is now considered canon in area the in areas of commutative algebra and algebraic geometry. Hilbert himself developed his famous "basis theorem" to solve the problem for the case of matrix groups acting on polynomials with complex coefficients. The problem also drew the attention of possibly one of the best female mathematicians of all time, Emmy Noether, who extended Hilbert's solution significantly. Unfortunately the general case has a negative answer by Nagata. During the talk, we will introduce the important class of rings, named after Noether, as well as a version of her original proof based on Hilbert's work and some conclude with some open questions.