DRP – Fall 2023

In The Symmetries of Things, Conway, Burgiel, and Goodman-Strauss take a non-algebraic approach to classifying spherical and planar patterns with their Magic Theorem. To show why the theorem works, the four fundamental pattern features–kaleidoscopes, gyrations, glides, and translations–will be introduced. The features define a pattern’s unique signature. Euler’s Map Theorem is used to assign a characteristic value to each pattern’s surface.

Signatures create orbifolds by identifying points in a pattern where the same symmetries occur. Applying the Map Theorem, the Magic Theorem assigns a cost to every fundamental feature in a pattern’s orbifold. The theorem’s results illustrate all possible spherical patterns. The results for planar patterns follow.

Today, cryptography is the basis of security for every message that we send to one another. The math encoding this important information employs modular arithmetic and the clever use of prime numbers. In this talk, we look at the basics of cryptography, the discrete log problem, and the ElGamal Key Cryptosystem. ElGamal is a public key cryptosystem that allows two people to communicate safely without establishing a key beforehand. Finally, we will talk about the Babystep-Giantstep Algorithm, and how it can be used to break ElGamal.

A flow on a directed graph, $\mathsf{G}$, is an edge labeling such that for any vertex, $\mathsf{v}$, the sum of the edges pointing towards $\mathsf{v}$ is equal to the sum of edges pointing away from $\mathsf{v}$. In this talk, I will give a basic introduction to flows and theorems that highlight relationships between algebraic concepts, flows, and colorings of graphs. In particular, I will be presenting Tutte’s flow conjectures, which have remained unproven for over fifty years.