How many ways can we write a valid sequence of $2n$ parentheses? How many rooted binary trees are there on $n$ vertices? How many up-right paths from $(0,0)$ to $(n,n)$ stay on or below the line $y=x$? In this class, we will explain what these questions mean and why they are all answered by a sequence called the Catalan Numbers.

Some things in math look true but are false, and some things in math look false but are true.

What's Esperanto? It's the most widely spoken invented language, actively spoken by around 200,000 people all over the world. It's really easy to learn! You'll learn more Esperanto in this hour than you'd learn German in ten hours. By the end of the class you'll be able to form basic sentences in Esperanto.

How many ways can we write a valid sequence of $$2n$$ parenthesis? How many rooted binary trees are there on $$n$$ vertices? How many up-right paths from $$(0,0)$$ to $$(n,n)$$ stay below the line $$y=x$$? In this class, we will explain what these questions mean and why they are all answered by a sequence called the Catalan Numbers.

This course introduces the infamous Fermat's Last Theorem (FLT), which remained unsolved for over 350 years despite its popularity among mathematicians. FLT claims that there are no nontrivial solutions to the equation $$x^n+y^n=z^n$$ for $$n\ge 3$$. We begin by covering the historical progress on special cases of $$n$$. We finish by introducing the concept of elliptic curves, and briefly covering the machinery that led to Andrew Wiles' proof of FLT in 1994.