When most of us hear "cryptography", we think of code-breaking, spies, or maybe even cryptocurrency. But even though these are the most obvious applications, there are many interesting real-world problems we can solve with the theory of cryptography. Consider this problem: everyone knows you need two officers with two different keys in order to launch a nuclear weapon. But in the 21st century, no one wants to carry a physical key. So the military goes out and buys a password-protected system to secure the weapons. Unfortunately, the system only requires a single password. How do we "split" or "share" the secret password in a way that both officers are needed to launch the weapon? Cryptographers call this problem "secret sharing" and with a little bit of math, we can implement it in a provably secure and correct way. We will learn about polynomials over finite fields, Gaussian elimination, the Shamir Secret Sharing scheme, and applications of secret sharing.

With faster computers, better algorithms, and innovation in AI, machine learning, deep learning, and quantum computing, it seems like computation can instantly solve almost any problem in logistics, mathematics, and even society. But what if some problems are simply too time-consuming to solve? Despite all the progress in computer science, we have failed to find fast algorithms for many interesting problems. However, some of these problems share an intriguing property: once you do find an answer, it is easy to verify. Does this help us solve them? Surprisingly, despite decades of work and a million dollar bounty, no one knows. This is the famous P vs NP problem. In practice, we generally assume P != NP, so if P = NP, the computational world would be turned on its head. Though some useful tasks would suddenly become easier, encryption would be impossible. Even mathematics itself might be at risk. Come learn the basics of computational hardness and the surprising utility of hard-to-answer questions.

Using just a compass and straightedge (unmarked ruler), can one trisect a given angle? Construct a square with the same area as a given circle? Construct a cube with double the volume as a given cube? These three compass and straightedge construction problems perplexed the ancient Greeks. As it turns out, these tasks are impossible! In this class, we will find out why this is the case and formalize what constructions are possible.