A Galois representation is a mathematical concept that connects number theory and algebra. It describes how the symmetries of solutions to polynomial equations, represented by the Galois group, can be expressed through linear transformations. These transformations act on vector spaces, allowing mathematicians to study the properties of numbers and their relationships. In particular, Galois representations are often used in the study of modular forms and elliptic curves. They provide a way to understand how these objects behave under the action of the Galois group of a number field, revealing deep connections between different areas of mathematics.