Galois Representations
Galois representations are mathematical structures that connect number theory and algebra. They provide a way to study the symmetries of solutions to polynomial equations by associating them with linear transformations. Specifically, these representations map elements of the Galois group—which describes how roots of polynomials can be permuted—to matrices over a field, allowing mathematicians to analyze the properties of these roots.
These representations are crucial in understanding the relationships between different fields, particularly in the context of modular forms and elliptic curves. They help in proving deep results, such as the Langlands program, which seeks to relate number theory and representation theory.