Proof theory is a branch of mathematical logic that focuses on the structure and nature of mathematical proofs. It studies how statements can be derived from axioms using formal rules, providing a framework to understand the validity of arguments in mathematics. In proof theory, different systems, such as natural deduction and sequent calculus, are used to represent proofs. These systems help mathematicians analyze the relationships between various logical statements and the consistency of mathematical theories, contributing to fields like computer science and philosophy.