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

Math Video Party!

Who makes your favorite math videos? In this course we'll scour the internet for the best math videos out there. You'll watch them during the week and then we'll discuss them in class - answering all your questions and going into even more depth on the topics presented.

Prerequisites: Instructor Approval

Calculus (Just the fun parts)

Calculus usually has a lengthy list of prerequisites, but we'll circumvent these using Desmos and Sage. Using these tools we'll discover and explore patterns that have fascinated some of the greatest mathematicians throughout history. Although we'll use modern computational tools, we'll approach the subject from a historical perspective, putting students in the shoes of Newton and Euler as they stumbled upon these patterns and seemingly paradoxical results. Topics include: Derivatives, Integrals, Infinite series, Power series and Differential equations.

Prerequisites: Students should have completed Algebra 1 but must not have begun Calculus.

Number Theory: Beyond the Integers!

Number theory began with questions about the natural numbers 1,2,3,... but these questions led to the study of other mathematical systems such as the integers mod n. The patterns in these other number systems were not only fascinating in their own right, but helped mathematicians answer questions about the natural numbers. In this follow up to our Introduction to Number Theory course we will continue to explore other number systems such as finite, quadratic and cyclotomic fields and their arithmetic. Along the way we'll also visit polynomial rings and p-adic numbers. We'll see that in many of these systems, we can ask the same kind of questions as in the natural numbers, and we can compare the behavior of these systems. We'll also see how the behavior in these systems sheds light on the behavior back in the natural numbers.

In this course 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: An introductory course in number theory including unique factorization and modular arithmetic. Some exposure to programming (e.g. 'if statements' and 'for loops').

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').

Linear Algebra

Linear transformations lie near the core of almost all of modern mathematics. A deep understanding of linear transformations and their algebra will give you conceptual tools you need for number theory, abstract algebra, geometry, analysis, and probability. Don’t rush through this subject or skip it to go to more “advanced” classes! In this class we will dive deep into the subject. Along the way we will point out direct connections to other branches of mathematics, science, and engineering. Computing tools like Octave and Sage will put rich and realistic examples at our fingertips, letting us build practical skills while seeing high-dimensional phenomena directly.

Prerequisites: Algebra 2. Mathematical maturity. Some exposure to programming (e.g. 'if statements' and 'for loops').

Introduction to Algebraic Number Theory

Is 2 a prime number? Well it depends on what number system you're working in! The ordinary integers? Gaussian integers? Eisenstein integers? In this advanced course we'll use python programming to explore the arithmetic of quadratic number fields. We'll use primary sources to explore how mathematicians stumbled upon the failure of unique factorization in these systems and how they salvaged this loss by discovering the theory of ideals. We’ll then see applications of the theory to concrete questions in number theory.

Prerequisites: An introductory course in number theory. An interest in python programming. Mathematical maturity.

Dynamics & Chaos

When you set a physical or mathematical system in motion, how might it evolve over time? Could it settle down to a steady-state, blow up, or exhibit some other kind of behavior? And if you wiggle the initial state of the system, how could that affect the outcome? You can find an example of such a system here.

This course provides an introduction to the theories of dynamical systems and chaos. We will use computational tools to study and make predictions about such systems, and then use mathematics to see if we can justify our predictions.

Prerequisites: Algebra 2. Mathematical maturity. Some exposure to programming (e.g. 'if statements' and 'for loops').

Discrete Mathematics

In this course we will explore a wide variety of discrete structures with connections to pure mathematics and computer science. Topics include combinatorics, graph theory, discrete probability and discrete calculus.

Prerequisites: Algebra 2. Mathematical maturity.

Fields and Galois Theory

In this course we’ll explore which numbers can be built out of others. We’ll find that an unusual kind of symmetry governs the answer to such questions. For example, we’ll see the role symmetry plays in finding the general solution to the quadratic, cubic and quartic equations, as well as revealing why it was so hard for mathematicians to find formulas for higher-degree equations. Most accounts of these topics rely upon ad-hoc ‘tricks’, but we’ll draw a straight line between these concrete calculations and the seemingly abstract ideas of Galois himself. Instead of just reading about these ideas, we'll use Sage and interactive tools to help us discover them for ourselves.