DRP – Spring 2023

The question of how to most securely encode private information has been present for millennia. Today’s powerful technology forces experts to find increasingly clever methods to ensure secure communication. In this talk, we will discuss the birth of modern public key cryptography and one of its most notable algorithms: the Diffie-Hellman key exchange, which allows for secure exchange of cryptographic keys over public channels. We will also investigate the underlying math that ensures its security, as well as potential methods which can be used to attack the algorithm.

RSA security systems are used throughout the internet for encryption. In this talk, we will discuss the modular arithmetic that makes these systems work, and explain why. Then, we will explore the $\mathsf{p} − 1$ attack, as well as how to find the large primes required to make these systems work.

In “The Non-Euclidean Symmetry of Escher’s Picture ‘Circle Limit III’ ” by H.S.M. Coxeter, Escher’s woodcut “Circle Limit III” is explored as much as a work of mathematics as a work of art. Through the colorful fish that fill the disc, we consider regular tessellations of the hyperbolic plane and groups of hyperbolic motions. The fish swim along arcs that meet the boundary of the disc at approximately 80$^\circ$, rather than 90$^\circ$ as Escher thought they should. We will show why this is the case, affirming Escher’s artistic integrity.

In this talk, we survey famous results on colorings and list colorings of graphs. Our main focus will be Galvin’s Theorem, which draws on the ideas of proper edge colorings, orientations and stable matchings.

What if there were a number greater than all the integers? Or a number bigger than zero, but less than all the positive reals? In this talk, we will answer these questions by looking at the surreal numbers, first created by John Conway. We will discuss the properties of this special well-ordered field and answer the above questions with constructions of both ordinal and infinitesimal numbers.