Mathematical research involves the exploration and study of mathematical concepts, theories, and problems. Researchers aim to discover new results, develop new methods, or provide deeper insights into existing mathematical ideas. This work can range from pure mathematics, which focuses on abstract concepts, to applied mathematics, which addresses real-world problems. Researchers often publish their findings in academic journals and collaborate with others in the field. They may also present their work at conferences, contributing to the broader mathematical community. Notable areas of research include algebra, geometry, and number theory, each with its own unique challenges and applications.