BRAINS (now with math!)

This summer, I want to go to each of the million biggest cities in the world. I want to save money though, so I need to know in what order I should go to these cities to minimize my total airplane ticket cost. Please help! Unfortunately, if you believe that the computational complexity classes P and NP are different, I am out of luck. There is no way I could possibly compute the best path in time. But luckily, I have some room in my budget - I'm willing to settle for any route whose cost is within 50% of the price of the best route. Can I quickly find a decent plan then? What if I need within 25%? Such is the domain of the PCP theorem, the most important theorem in computational complexity proved in the last 30 years. We will talk about approximate solutions to hard (NP-complete) problems, and our best ways of showing that they don't exist.

This class will focus on the development of each individuals students' writing style. Specifically, we will work on making our writing pieces clear, concise and effective at communicating their intended meaning(s). To facilitate the development of our own writing, we will read and analyze essays and short stories of notable authors. We will practice editing our own writing and the writing of other students. Finally, we will cover technical aspects of writing including formatting, annotations and citation styles. This class will be writing intensive, students should expect to write at least one double-spaced page and read one short piece in preparation for each meeting.

WHAT IS THE SHAPE OF THE UNIVERSE? IN 1982, MATHEMATICIAN WILLIAM PAUL THURSTON REALIZED THAT ALL THREE DIMENSIONAL SHAPES CAN BE BUILT OUT OF JUST 8 BASIC GEOMETRIES. WHAT CAN THIS DO FOR US? IN 2010, A MAJOR FURTHER BREAKTHROUGH HAPPENED. FASHION DESIGNER DAI FUJIWARA DISCOVERED THAT HE COULD MAKE CLOTHES OUT OF THESE GEOMETRIES, FINALLY GIVING US ACCESS TO THE TRUE NATURE OF THE UNIVERSE. WHERE CAN I LEARN MORE? WWW.AMS.ORG/PUBLICOUTREACH/AMS-NEWS/RELEASES/THURSTON-MIYAKE AND ALSO THIS CLASS

Learn about the chemistry of hydrogen!

Learn about the chemistry of helium!

Learn about the chemistry of lithium!

Learn about the chemistry of beryllium!

Learn about the chemistry of boron!

Learn about the chemistry of carbon!

Learn about the chemistry of nitrogen!

Learn about the chemistry of oxygen!

Learn about the chemistry of fluorine!

A Mobius transformation of the complex plane sends z to $$\frac{az+b}{cz+d}.$$ Any collection of three points can be sent to any other collection of three points by a Mobius transformation. Two collections of four points can be sent to each other only if they have the same cross ratio. What happens with more points?

There are the real numbers, the complex numbers, the quaternions, and finally, the octonions. All these systems have multiplication and inverses. We'll show that they're the only such systems using topology.

R C Q O Guess what the above letters mean. Now can you guess what these mean? E6,E7,E8 F4, G2 Well, we got bored of rotations, so we're going to use division algebras to give us new weird kinds of symmetries. Anyways, can you guess the meaning of this square? A1 A2 C3 F4 A2 A2xA2 A5 E6 C3 A5 D6 E7 F4 E6 E7 E8

Three dimensions. Rotations. A sphere. Ten dimensions. Rotations. Two OGres. We are going to cut the OGres into pieces. By doing so, we will get a good understanding of the most symmetrical kind of geometry, the geometry of OGres. Maybe even affine OGres. AKA: "What do really symmetrical high-dimensional objects look like?"

The goal of this class is to expose students to a topic on the cutting edge of algebra. String theorists predict the existence of an 6-dimensional physical theory called Theory X (that's actually the name.) It turns out that Theory X leads to lots of interesting mathematics (very relevant here is the fact that 6=2+4.) Come learn about it.

A quiver is one of the most accessible objects which have an interesting "representation theory". A quiver is just a graph where each edge has a direction. Imagine that you have a vector space for each vertex (if you don't know what a vector space is, you'll be taught) and a map for each edge. What happens????

There is 1 line through 2 points. There is 1 conic through 5 points. There is 1 conic tangent to 5 lines. Boring, huh? How many conics are tangent to 5 conics? (3264.) If you have a degree three polynomial in three dimensional space and you look at the surface where it's zero, how many lines lie on this surface? (27.) We will solve these problems and then explain how this is related to something called "quantum cohomology."

A quadratic form is just a degree 2 polynomial (in more than one variable). What interesting things can you say about these? Well...