Since ancient times, mathematicians have been incredibly intrigued by prime numbers: integers greater than or equal to two that are only divisible by themselves and one. At first, there are many prime numbers, and they are packed closely together: 2, 3, 5, 7, 11, 13, and so on. We know that later, however, the primes become much less closely packed, so much that we have to use supercomputers to find very large primes! The study of the distribution of prime numbers has as a result been a very involved subject of study in mathematics, and one which is used in applied fields like cryptography. We'll show, using only high-school level techniques, how an interesting function called the Riemann Zeta function can tell us many useful things about the distribution of prime numbers. We will also mention connections to the Riemann hypothesis, one of the greatest unsolved problems in math today.

The universe is pretty much the oldest thing around! We'll discuss the various efforts historically to derive an accurate formula for its age before learning the modern method directly from first principles, and computing [the officially recognized value] using values of observables from the Planck 2013 satellite. Along the way we'll also derive a fundamental equation in physics (the Friedmann equation) and learn how cosmology uses the tools of both quantum mechanics and general relativity (no prior knowledge of either necessary).

The universe is pretty much the oldest thing around! We'll discuss the various efforts historically to derive an accurate formula for its age before learning the modern method directly from first principles, and computing [the officially recognized value] using values of observables from the Planck 2013 satellite. Along the way we'll also derive a fundamental equation in physics (the Friedmann equation) and learn how cosmology uses the tools of both quantum mechanics and general relativity (no prior knowledge of either necessary).

Since ancient times, mathematicians have been incredibly intrigued by prime numbers: integers greater than or equal to two that are only divisible by themselves and one. At first, there are many prime numbers, and they are packed closely together: 2,3,5,7,11,13, and so on. We know that later, however, the primes become much less closely packed, so much that we have to use supercomputers to find very large primes! The study of the distribution of prime numbers has as a result been a very involved subject of study in mathematics, and one which is used in applied fields like cryptography. We'll show, using only high-school level techniques, how an interesting function called the Riemann Zeta function can tell us many useful things about the distribution of prime numbers. We will also mention connections to the Riemann hypothesis, one of the greatest unsolved problems in math today. This class is similar in spirit to M8501, but covers a more focused topic and will not require previous exposure to abstract math.

"Suppose we throw a goat into a black hole -- well, a gas of goats, and each has a blinking light on its head..." ~Allan Adams, MIT professor This class will be a survey of interesting phenomena and counterintuitive concepts regarding black holes and other interesting ideas in relativity and cosmology. Mathematics, like seasoning, will be used to taste. Most likely this will include basic concepts in special and general relativity, i.e. the metric tensor, Schwarzschild metric, and invariant interval, and their applications to the physics of black holes.

Maybe you don't like math in school. Maybe you think math in school is pretty okay but haven't seen much more than that. Maybe you have an intimidatingly clever friend who knows a thing or two about * how many prime numbers there are (hint: a lot), * how many real numbers there are (hint: even more), * figuring out the day of the week an arbitrary date was on, * simple-sounding math questions that literally no computer ever can or will answer, * saving infinitely many prisoners from guessing their hat color incorrectly by using some controversial set theory, and maybe even a few other things. Maybe there's a Splash class you can take that will introduce some of that stuff and give you a taste of what math outside of high school is like. Maybe you should take it. Maybe it will be fun.

This course prepares students to take the AP Physics C exam in the spring.

The Fundamental Theorem of Calculus. The Fundamental Theorem of Calculus for Line Integrals. Green's Theorem. Stokes' Theorem. Gauss' (Divergence) Theorem. ALL OF THESE ARE THE SAME THING. Using the language of differential forms, I'll show how all of these theorems are unified within a single elegant framework. Each is a really a special case of a more overarching theorem, also called Stokes' Theorem.

Conservation of Energy? Conservation of Momentum? Where did all of these arbitrary laws come from and why are they true? Starting from empirical principles, I'll show that the underlying symmetry of the universe enforces conservation laws. If we have time, I'll show how we can use these symmetries to prove that a field like the Higgs field must exist.