Introduction to Number Theory

It’s easy to come up with complicated questions that are hard to answer (e.g. What are the integer solutions to 54x² + 23xy - 47y - 2^y = 0 ?). It's also easy to come up with simple questions that are easy to answer (e.g. What’s 1+1? What's the 100th even number?). But the rarity is a simple question that is hard to answer. The natural numbers 1,2,3,... are some of the simplest mathematical objects one can imagine, yet simple questions about them have remained unanswered for millennia, and attempts to answer these questions have given birth to whole new fields of mathematics. In this course we will explore these questions and seek out our own!

In order to aid our discovery, we’ll use computational tools such as Sage to quickly perform experiments, allowing us to formulate and test our conjectures. We’ll then work together to try and explain what we are observing.

Prerequisites: Instructor Approval

A Computational Tour of Group Theory

How do you think budding geologists fell in love with their subject? By reading about rocks in a textbook? Their love probably began when they found themselves playing in the dirt, picking up the objects around them, marveling at these objects and asking questions about them.

In this course we'll play in the mathematical dirt, so to speak, with groups. This process would usually be prohibitively time-consuming, but we'll use computational tools such as Sage to quickly survey a wide swath of specimens. Like in any population, we'll be on the lookout for typical behavior among the specimens we find as well as unusual behavior. And we'll find that the theory of groups will help us make sense of what we're observing.

Prerequisites: Mathematical maturity. Some exposure to programming (e.g. 'if statements' and 'for loops').

The Theory of Partitions

How many ways can you break a whole number into parts? For example, the number 3 can be broken into 1+1+1, 1+2 and 3 itself (order doesn't matter). We'll find such a simple question gave rise to deep mathematics that captured the imagination of mathematicians from Ramanujan to Ken Ono! In this discovery-based course we’ll use computational tools such as Sage to quickly perform experiments, allowing us to formulate and test our conjectures. We'll then use a blend of combinatorics, number theory, algebra and more to try and prove them!

Prerequisites: Instructor Approval. Some exposure to programming (e.g. 'if statements' and 'for loops').