Symplectic geometry is a branch of differential geometry that has its roots in classical mechanics (specifically Hamiltonian mechanics) because of its usefulness in describing the phase space of a mechanical system. In this class, we will explain and motivate the basic notions of symplectic geometry, with a view towards the relevant physics. Time permitting, we will also cover some basic theory of symplectic embedding problems.

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.

A classic joke goes that a topologist is someone who can't tell a coffee cup apart from a donut. We'll talk about what it means for two spaces to be "the same" to a topologist and how we can tell apart spaces that are actually different. We'll study tools like the Euler characteristic and the fundamental group and see how they can be used to study spaces. Some cool applications of topology that I'll discuss include (but are not limited to!): If you drop a map on the ground, some point on the map is on top of the point it represents. There is some point on the Earth with the same temperature and humidity as the point directly opposite it. Furthermore, there is a point on the Earth with no wind.

Mathematics isn't always as it seems. This class will be a melange of mathematical statements that seem utterly false but are indeed true, and others that seem so true but just aren't so. Don't worry, we won't have any normie examples like Monty Hall or Banach-Tarski.

A classic joke goes that a topologist is someone who can't tell a coffee cup apart from a donut. We'll talk about what it means for two spaces to be "the same" to a topologist and how we can tell apart spaces that are actually different. We'll study tools like the Euler characteristic and the fundamental group and see how they can be used to study spaces. Some cool applications of topology that I'll discuss include (but are not limited to!): If you drop a map on the ground, some point on the map is on top of the point it represents. There is some point on the Earth with the same temperature and humidity as the point directly opposite it. Furthermore, there is a point on the Earth with no wind.

Classical number theory is one of the oldest and most intricate branches of math out there. Diophantine equations (named after the Greek mathematician Diophantus of Alexandria) involve finding when equations have integer solutions. In spite of their apparent simplicity, they can be devilishly hard- Fermat's Last Theorem was a Diophantine equation that remained unsolved for over 350 years! Although solving Diophantine equations will be one motivation for the number theory we develop, the entire class will have a view to applications to other branches of math and, yes, even the real world. To whet your appetite, here are a few possible topics that we might discuss: Is there a way to systematically generate all Pythagorean triplets? Which numbers can be written as the sum of two squares? How is number theory used in cryptography?

"If you can state a problem in mathematics that's unsolved and over 100 years old, it is probably a problem in number theory" (Erdős). We will describe the Riemann zeta function in enough detail to give a precise statement of the Riemann Hypothesis- a conjecture about the zeroes of this function that has confounded mathematicians since it was posed in 1859. Although the Riemann Hypothesis, strictly speaking, is not in the realm of number theory, it is interesting mainly because of the implications it has to the distribution of the prime numbers. These connections to number theory will be discussed, time-permitting.

Number theory -- the study of integers and their properties -- was long thought to be a subject with little real-world relevance. Hardy, in his essay, A Mathematician's apology, is thankful that his work in number theory cannot be used for malicious purposes- since it cannot be used at all! Some years later the situation is different, as number theory is now integral in cryptography, the theory of encrypting and decrypting messages so that the intended recipients, and only the intended recipients, can read them. Security of computer systems in one of the biggest issues in the modern era. Learn how one of the oldest problems in mathematics, factoring integers, can be used to create encryption schemes powerful enough to protect sensitive information in a digitalized world. We will use both proof and implementation to demonstrate the goals and techniques of cryptography and a commonly used cipher.