Crystallographic Groups in Two Dimensions
In this project we will work through John Conway's colorful book The Symmetries of Things to uncover the underlying properties of wallpaper patterns in two dimensions. Although this project is light on symbolic math, we will use Conway's ingenuous and highly rigorous system to gain a deep, intuitive understanding of mathematical symmetries. We will prove a classification theorem, showing that there are exactly 17 two-dimensional crystallographic groups. We will also examine wallpaper patters on the sphere, and relate these to the classification of planar patterns through stereographic projection. Along the way we will gain familiarity with mathematical tools such as triangulation of surfaces, the Euler Characteristic, orbifolds, algebraic groups and their actions, and topological quotient spaces. This project is self contained and requires very little prior mathematical knowledge. Some familiarity with trigonometry, vectors and matrix multiplication is helpful, but is in no way required.
Introduction to Mathematical Analysis
This project will give the student a first look at formal, proof-based mathematics and will stress problem solving and creativity. We will start with the foundations of set theory and construct the rational, real and complex numbers, gaining a deep understanding of their properties along the way. Then we will solve problems in more general metric spaces, where we will explore sequences and their convergence, and allowing us to tackle basic topological concepts such as openness, compactness, and connectedness. Time permitting, we will discuss continuity in a metric space, as well as in more general, non-metric topological spaces, and we will play around with a bunch of continuous functions to build intuition. The student will explore many hands-on examples and solve problems that illustrate each concept. The final project will be a lecture, given by the student, where the student presents the proof of a theorem in real analysis and gives examples illustrating its use.