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Cluster 6
Mathematics

 

 

Introduction

This cluster is designed to introduce students with a strong interest in mathematics to several advanced topics. These topics would ordinarily be studied at the advanced undergraduate 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

Enumerative combinatorics provides a way to count in complicated mathematical settings, and we will learn several important counting techniques in this course. In addition, we will learn some related ideas in elementary probability and number theory.  The following is an example of a combinatorics problem that we will encounter in our four-week journey through this interesting area of mathematics:
If a teacher returns a test to her class of 10 students at random, what is the probability that no student gets his or her own test?

 

Supplementary Courses

 

 

Knots, Links and Topology (2 Weeks) 

 

From knotted strands of DNA to tangled necklaces, knots and links arise in numerous contexts. 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 general 3-dimensional spaces. For example, we'll examine whether or not our universe is flat. We now know that the earth we live on is not flat, but what about the 3-dimensional universe we live in? If we send a rocket probe into space, programmed to go "straight", will it eventually come back to where it started, as happens if you fly “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. What possibilities are there for 3-dimensional spaces?   Is it possible that we can reach the “edge” of the universe? We will learn the basic notions of topology, the mathematical theory that allows us to study problems of this kind.

   

Introduction to Graph Theory (2 Weeks)

 

A graph is a collection of points, called vertices, connected by edges.    Using graphs to describe information has applications from scheduling tournaments to cryptography to understanding the structure of the internet. We’ll study the basics of graph theory, including graph coloring problems, Euler’s formula for graphs on surfaces, and graph planarity.