Mathematical invariants are properties or quantities that remain unchanged under certain transformations or operations. For example, in geometry, the area of a shape is an invariant when it is transformed through rotation or translation. Invariants help mathematicians understand the essential characteristics of objects, regardless of how they are manipulated. Invariants are crucial in various fields, including algebra, topology, and physics. They provide a way to classify and compare different mathematical structures. For instance, in group theory, the order of a group is an invariant that helps identify its properties, while in differential equations, conserved quantities like energy serve as invariants that remain constant over time.