In 1900, Hilbert delivered 23 problems in the International Congress. The problem 14 in the list of 23 problems is known as the Hilbert's $14^{th}$ problem. The problem arises out of the classical invariant Geometry and leads to some subtle interactions among geometry and representation theory. A special case of the problem can be stated as follows: let $K$ be a field. Let $G$ be a subgroup of $GL_n(K)$ acting on the polynomial ring $A=K[X_1, X_2,...,X_n]$. Let $A^G$ be the ring of elements of $K[X_1, X_2,...,X_n]$ invariant under $G$. Is $A^G$ finitely generated over $K$? The first counterexamples were presented by Nagata in 1958. However, the problem is affirmative in some of the special cases. Mumford proved when $G$ is a linearly reductive algebraic group and $K= \mathbb{C}$. In the talk, we shall study Mumford's proof. Moreover, the talk will be also a brief introduction of the Geometric Invariant Theory (GIT).
