This course is the equivalent of a full-year, honors-level course in abstract mathematics. It is designed for highly motivated students who have completed Precalculus and are eager to challenge themselves beyond the standard curriculum. Students will explore a diverse range of advanced topics, including number theory, quaternions, abstract algebra, graph theory, and game theory. Throughout the course, emphasis will be placed on developing students’ ability to construct clear, rigorous mathematical proofs while gaining exposure to higher-level mathematical thinking. In addition, the course aims to connect these areas of pure mathematics to real-world applications, demonstrating how pursuing mathematics for its own sake can lead to meaningful impact. Applications will be drawn from fields such as computer science, political science, philosophy, and others. The course provides both a strong foundation in proof-writing for mathematics competitions and an introduction to topics beyond the traditional applied curriculum. Daily homework will be assigned, and weekly in-class summative assessments will be administered. This course will meet Monday through Friday from 8:00 am to 12:00 pm from June 8 until July 30, with no class during the week of June 15. Each week of the course is generally self-contained, so students who miss no more than one week are encouraged to enroll and coordinate make-up work with the instructor. However, students who plan to miss intervals of time in multiple different weeks will likely face difficulty in completing the course. Questions about course pre-registration procedures or content can be directed to Mr. Henry Erdman at rerdman@loyolablakefield.org.