Mathematics Projects

Title: Combinatorial Magic and Geometry

 

Combinatorics is the mathematics of methodically counting. This includes basics such as permutations and combination and inclusion-exclusion. A typical (and now classic) geometric question is “how many ways can non-intersecting diagonals be drawn into a decagon so that all the resulting regions are triangular?” Some of the combinatorial principles we investigate will be illustrated with card tricks. Despite the whimsical-sounding nature of some of the problems, combinatorics has important applications in computer science and other fields. Some numerical investigations will be done with a computer – students will learn how to use SAGE, a computer algebra system that uses the Python language syntax. Students will choose their main research projects from several topics presented during our early sessions.

This group will concentrate on that part “high school level” mathematics that is often left out of high school mathematics but is often necessary to do well in mathematics competitions. Topics will be include parts of combinatorics, modular arithmetic, geometry, inequalities, and developing computer algorithms for cryptography and Monte Carlo Methods – students will learn how to use SAGE, a computer algebra system that uses the Python language syntax. This will culminate in the investigation of problems that the students themselves have developed.

 

Title: Chaos Theory and Fractal Geometry

Explore the world of chaos by looking at discrete models that erupt with unpredictable behavior. This chaotic behavior is commonly associated with weather patterns and stock market volatility, but we consider models for interactions in insect populations, the spread of disease, for determining earthquake fault lines, and models for tumor growth.  We will also learn how discrete models can give rise to fractal images.