Cluster 6
Mathematics

• Instructors: Lawrence Marx, Abigail Thompson & Kurt Kreith
• Prerequesites: Algebra II
• Typical Field Trips: Exploratorium, Lawrence Hall of Science & Electronic Arts
• This is a FIRST CHOICE option only.

Introduction

This cluster is designed to introduce students with a strong interest in mathematics to several different advanced topics. Many of these topics would ordinarily only be seen at the graduate level, but all lend themselves to an introductory course at the high school level. No prior experience in any of these topics is expected, but enthusiasm for and interest in mathematics is essential.

Core Course (4 Weeks)

Combinatorics

Combinatorics provides a sophisticated way to count in complicated mathematical settings. Here are a couple of examples of combinatorics problems that we will encounter in our four-week tour of this elegant area of mathematics: 1. Suppose there are six people in a room. Show that either there are three of them all of whom know each other or there are three of them none of whom know each other. 2. Given six girls and six hats, how many ways are there for each girl to get a hat that doesn't belong to her?

Supplementary Courses (2 Weeks Each)

Knots, Links and the Topology of Space

From knotted strands of DNA, to tangled necklaces, to the basis of String Theory, knots and links arise in many different applications. We’ll look at the mathematical theory of knots, and learn what knot invariants can tell us about them. We’ll connect knot theory to the study of all 3-dimensional spaces. For example, we'll examine whether or not our universe is flat. We thought the earth was flat for a long time... what about the 3-dimensional universe we live in? If we send a rocket off into space programmed to go "straight", will it eventually come back to where it started, like what happens if you go “west” long enough starting at a point on the equator? We'll look at some of the possibilities by studying what is known about the geometry and topology of space.

Some Number Theory - With a Computer At Your Side

"What is 14÷3?" In reconciling possible answers to such questions we will encounter a novel form of "long division",an important theorem due to Fermat, and some powerful tools for the encryption of data - including the famous RSA algorithm. Spreadsheets will be used to harness computer technology in the implementation of concepts so developed.

Modified 2009-02-05T22:37:51Z