Program I topics are introduced through five motivating problems such as limitations of straight-edge and compass constructions, classification of patterns in two dimensions, error-correcting codes, cryptography, and the analysis of symmetry in structures.
The mathematics central to solving these problems comes from the areas of abstract algebra and number theory. Abstract algebra originated in the early part of the 19th century through the study of polynomial equations. This branch of mathematics lies at the core of many areas of modern mathematical research. Number theory concerns properties of the integers, and has its origins in ancient mathematics. Number theory remains a very active field of study with interesting open problems and important applications in computer science.
Recommended Prerequisite Mathematics Experience for Program I:
Students applying for Program I should have experience writing and reading mathematical proofs, and strong high school geometry and algebra mastery. SUMaC Program I applicants should be comfortable with:
- Proofs by induction, contradiction, contrapositive, and more.
- Logic used in mathematics such as basic logical symbols and their meanings like if, then, or, and, etc.
- Notation for subsets, supersets, and intersections.
Students accepted to Program I have typically studied number theory, and are comfortable with modular arithmetic and some basic theoretical results involving modular arithmetic. Prior participation in mathematics competitions or contests is not required.