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CLUSTER VI. Mathematics

Instructors: Kurt Kreith, Patrice Koehl, Monica Vazirani, Duane Kouba, and Lawrence Marx
Prerequisites: Algebra II
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. Therefore this cluster available as a FIRST CHOICE option only.

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 will start this course off on a two-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?

Mathematics lab activity

SUPPLEMENTARY COURSE A1 (2 WEEKS)
Introduction to Discrete Dynamical Systems
Much of the importance of calculus is rooted in its mechanisms for modeling change. The advent of computer technology has led to alternative ways of modeling change, ones that are rooted in algebra and involve surprising phenomena such as "cycles and chaos". Starting with problems from algebra, we will use spreadsheets to develop mathematical techniques that are in some ways more powerful than their calculus-based counterparts. These will be applied to issues of global change, including energy flows, population dynamics, and human migration.

SUPPLEMENTARY COURSE B1 (1 WEEK)
Biogeometry
Most living organisms are complex assemblies of cells, the building blocks for life. Each cell can be seen as a small chemical factory, involving thousands of different players with a large range of size and function. Among them biological macro-molecules hold a special place. These usually large molecules serve as storage for the genetic information (the nucleic acids such as DNA and RNA), and as key actors of cellular functions (the proteins). As the function of these molecules is directly related to their structure and shape, we have seen recently the emergence of a new partnership between mathematics, computer science and biology, namely bio-geometry. In this course we will show how classical and advanced geometric techniques are applied to the study of macromolecules, and more generally to biological process. Our goal in this class is to bring awareness to the students of the critical need of interdisciplinary approaches to study biological problems, focusing on mathematics.

SUPPLEMENTARY COURSE B2 (1 WEEK)
Symmetry
Symmetry is all around us: in nature, art, architecture, chemistry, music, and more. The petals of a flower, a bee's honeycomb, a sudoku puzzle, a mandala, a water molecule, even a mathematical equation; all exhibit beautiful symmetries. Using mathematics we can describe those symmetries, and more importantly we can put them "to work" for us. What if you wanted to count how many different sudoku there are? Listing them all would take a LONG time, but taking advantage of their symmetry, we can write down a formula that counts them. In this course, we will use symmetry to count these and other objects, such as "colorings" of regular polyhedra.