ESP Biography

Splash is run by students --- undergraduates and graduates at Stanford University. Beyond Stanford, there are Splash programs running at over a dozen universities nationwide and when you go to college, wherever you go to college, you have the opportunity to start another Splash yourself. Come learn about all of the intricacies that go into running a massive program like Splash and find out how you can do it yourself.

Addition and multiplication seem like such basic ideas. You can add numbers, and you can multiply numbers, and that’s all there is. But what if you could add or multiply things that *aren't* numbers? This fundamental mathematical generalization opens up a new world of much richer ideas, and uncovers the truth about algebra. We’re going to study ring theory, one branch of algebra that investigates what happens with a set in which you can add and multiply. We’ll focus on polynomial rings while developing ring theory in general. This class is going to be extremely fast and abstract, so be prepared for quite a ride if you come!

"If the Aborigine drafted an I.Q. test, all of Western civilization would presumably flunk it," wrote anthropologist Stanley Garn. What is intelligence, really? Can we measure it? If so, what does it tell us about the human mind? Is it something that's born into us by our genes, or does it depend on how we're raised? What makes someone smart? Not all of these questions have been answered, but there's some pretty good progress towards understanding them. Come discover just what it means (or doesn't mean) to be "smart".

Deep mathematics comes from taking simple concepts and generalizing them. Maybe you think you understand the idea of distance. Generalize it, and you get a “metric space,” a new object that satisfies all of the basic properties of distance but does far more interesting things than happen in our world. With metric spaces, we can understand the idea of continuous functions more deeply; we can start to understand the mathematical field of topology; we can prove the existence and uniqueness of solutions to differential equations; and much more. This class will offer a fast-paced introduction to the theory of metric spaces and a look at what you can do with them.

Imagine two teams of tennis players, the Avengers and the Bobcats. When every Avenger plays every Bobcat, the Avengers win more games. Now each gets a new player, and the Avengers' new player is better than the Bobcats new player... but now when every Avenger plays every Bobcat, the Bobcats win more games. How did that happen? Come to this class and explore three dice A, B, and C so that A usually beats B, B usually beats C, and C usually beats A. Discover why sometimes you should choose to play against a stronger opponent more than a weaker one. Most importantly, learn how to think clearly about difficult problems.

The complex numbers are an amazing mathematical object. But they are also an amazingly hard space to deal with. You can’t define a consistent square root, for example – the square root has to take on two different values. And taking logarithms is even worse: a logarithm has infinitely many values to it! We’re going to study the complex numbers and uncover how they really work. Our exploits will take us through what it means to take the derivative of a complex-valued function, on to a bit about integration, and finally talking about Riemann surfaces. Along the way, I’ll mention interesting things that come up such as the Riemann hypothesis.

Think you’re perfectly logical? Think that you see everything around you? That you remember things just how they happened? Turns out, you don’t. We’ll see just how your brain doesn’t work the way you think it does. It misleads you. It takes shortcuts, and tells you things that aren’t true. Be aware of where your brain goes wrong, and you’ll be smarter, better able to avoid being misled, and more aware of what’s around you.

On June 13, 2007, the New York Times reported that New York City students had made huge gains in math: as many as 11% more were passing the state math exam than the year previously. Does that mean that students had really gotten that much better in one year? It has been found that countries that use fluoride in their drinking water have a higher cancer rate than other nations. Should we stop using fluoride in our water? It has been reported in the media and elsewhere that 150,000 young American women die of anorexia each year. (I’ll give this one away: only about 60,000 women under the age of 50 die in the US at all each year, making this statistic totally impossible.) Sometimes the math and numbers scare people. Come see the many subtleties of statistics, and get a step closer to being able to discriminate the good from the lies.

In 1874, the mathematician Georg Cantor first came up with the profound ideas that led to "transfinite numbers." His insights allowed mathematicians to look at precisely what infinity means, to work with it, to understand exactly what they can do with this improbable concept. Now we can answer questions such as "when are two infinite collections of objects the same size?" We can understand how to compare the infinite set consisting of all integers with the infinite set consisting of all rational numbers (all fractions). And we can determine just how many sizes of infinity there are. Be prepared to have all your preconceptions thrown out the window in a challenging math class.

This class is about several different concepts in mathematics, and how they interact to produce some really stunning results. It's about the power of generalization. And it all starts with one simple question. What is distance? We'll ask what the most important ideas about the notion of "distance" are, and then find a way to generalize them to places far different than just the distance between two points in space. This will lead us to define mathematical objects called "metric spaces," sets of points where we can tell how far apart two points are, but nothing else. Yet even with just that --- just a notion of distance --- we'll be able to come up with a huge host of results, including, finally, the idea of "compactness," one of the most fundamental notions in mathematics. These ideas are extremely abstract, and you should come prepared for a very difficult math class. However, when we're done, we'll be able to prove a truly amazing result: every polynomial has a root in the complex numbers. With extra time, we'll use our results to discuss precisely the convergence of sequences and infinite sums.

Is math just memorization to you? Do you struggle through tests by memorizing formulas and hoping there are enough clues to know which number to plug in for which variable? Do you hate the word "variable?" This class may or may not be the cure, but it will show you a new way of thinking about mathematics. We'll see some of the most beautiful --- yes, beautiful --- things about math. Maybe you don't think that math *can* be beautiful. But I think it is, because it gives you insight into the workings of the human mind, much like a good piece of art. Come see why.

What are numbers, really? I mean, what *are* they? As children, we were taught how to count, as if numbers had always been there and were obvious. When you got to fractions, well, those were supposed to be clear too. And then real numbers? $$\pi$$? It was always brushed under the rug... it's just some weird decimal that goes on forever, right? Well, you can't prove anything about numbers if you don't know what they really are. How do we know that mathematical constructions actually work? What basis tells us that even something as simple as addition makes sense --- how do you even define it? What could it possibly mean to take something like $$\pi^{\sqrt{2}}$$? Well, the numbers can be built out of something much, much simpler. You can work your way right up from almost nothing to the full complexity of the real numbers. Come and find out how a mathematician thinks about a concept you might have thought was simple. This will be a very challenging course, but not like the mathematics you see in school: it won't be about memorizing formulas or lots of calculations. This class is for people who like and are good at dealing with abstract concepts and logical deduction.

Join us as we solve some of the deepest and most challenging puzzles around. These puzzles, seen in competitions such as the MIT Mystery Hunt, require careful analysis and deep logic, and they're often given without any instructions! We'll go through a sample of these puzzles and try to work through them together. Stretch your brain and grow your skills for finding deep patterns, examining open-ended questions, and pulling out solutions without nearly enough information.