## cSplash 2006

Computer Modeling of ClimatePredicting the earth's future climate represents an enormous mathematical and scientific challenge, and one with profound societal ramifications. In this presentation we focus on one particular aspect of climate change, namely that of global sea level rise. Computer models that constitute the basic tool for future prediction of sea level are presented and discussed. The key areas of ongoing research are highlighted. Traffic flow, rivers and oceansThe same mathematical framework: systems of conservation laws, describes a number of physical scenarios, including the flow of cars through roads, and of water through rivers and oceans. Through this unifying mathematical theme, we'll find analogies between traffic jams, river bores and tsunamis. The Reeb Foliation of the 3-SphereFirst, you get a circle. Go up a dimension and you get a "normal" sphere. Go up another dimension and you get the 3-sphere. This is a really interesting ! object: it has to sit inside four dimensions, because it doesn't fit in three-dimensional space, and it has a number of really interesting properties. We're going to study those properties, first by figuring out exactly what this 3-sphere thing is, and then by analyzing it by taking a "foliation." If that doesn't make sense, don't worry about it --- we'll go over it in class. But if you want to start to visualize things in four dimensions, this is a great class to do so. Proof without WordsDid you know many of the mathematical results can be proved without even writing a word? This class will be an introductory session to a different method of proving & approaching mathematical results; visually, those that do not involve any word comments in the proof. We will study common Mathematical results along with their visual proofs and study how various results could be represented visually. We will also attempt to use this technique to solve mathematical problems. The topics covered will include high school algebra, calculus, geometry & combinatorics. An Introduction to Financial MathematicsFinancial Products, so-called derivatives, have been created decades ago; however, it has only been a few years that they have efficiently entered the market. They are now a fast developing breed, also helped by the exponential growth of powerful computers. In this class, we will see different sorts of financial products, starting with an introduction to the market, to the need of derivatives, and then some more complex products and ways to give them a fair price. Ask the mathematician!Do you have a question about math, but never knew who to ask? Here is your chance! A real, live, mathematician will be here to answer all your questions. Ask him anything you want: about math, what it's like to be a mathematician, the purpose of math, and he even knows things about astrophysics, earth science, chemistry, and molecular biology. There is no such thing as a silly question! If nobody has a question, then he will ask YOU! |
The Math Behind GoogleEver wonder how Google knows which web sites are better than others? Well, part of the recipe is a company secret, but a very important aspect of Google search is the idea of PageRank, named after Larry Page (co-founder of the company). We start by introducing elementary graph theory, which is one very useful way to model the internet. One way to think about PageRank is as a random walk on a graph. Next we will see how to combine this idea with standard matrix multiplication. The result is a huge system of simultaneous equations, but we can't solve it in the usual way! This is because the variables to solve for are on both sides of the equation -- so how can we solve it? We'll see a very clever algorithm which might surprise you with its simplicity and effectiveness. And in the end, the solution to this crazy equation gives us our ranking of all the net's web pages! Graph Theory and the DonutDraw five dots on a page. Can you connect them without crossing any lines? We'll (rigorously) solve this puzzle --and answer lots of other questions about dots and lines-- in this 1-hour investigation into Topological Graph Theory. And the answers will not be limited to the plane: can you connect the same five dots if they're sprinkled on a donut instead of drawn on a piece of paper? You'll come away with a new understanding of the old joke about a mathematician being "someone who can't tell the difference between a donut and a coffee cup." Ordinary Differential EquationsCalculus is the foundation needed for solving all kinds of problems in science. This is because many problems in science can be stated as differential equations. In this class, we'll solve a few examples of differential equations. The examples might be population growth, electric circuits, mixing chemicals, or the pendulum. (With ordinary differential equations, we deal with functions of 1 variable, such as f(x). With partial differential equations, the function depends on more than 1 variable, such as f(x,t). We'll see why they're called partial differential equations in another class called "Partial Differential Equations".) So, Why is Einstein So Cool?You always heard that Einstein is, almost undoubtedly, the most influential physicist of the 20th century. But why is that so? What did he do that was so novel? In this talk I will try to enlighten the processes that led him to propose ultimately his Theory of General Relativity. Starting with why he postulated that the light has a fixed speed, with immediate result that our world is 4 dimensional, I will then attempt to explain the philosophical ideas that led him to think our world was in fact curved. I will give different examples (black holes, worm holes, twins paradox, etc.) of the extraordinary consequences of his theory. Finding Order in Chaos -- An Introduction to Ramsey TheoryTake a grid -- a very very big grid. Now color all the points red, green, blue or purple. Then I can always find (no matter how you color it) a square in your grid, where all the corners have the same color. In fact, not just a square, but any reasonable shape you want. This type of result -- no matter how you do the coloring, I can find something very ordered inside it (a square, say) is an example of Ramsey theory. We will discuss this and other examples, including, maybe, tic-tac-toe in high dimensions, or the game of Set. |