EXPOSITIO

The Fundamental Theorem of Calculus, Green's theorem, the classical Stokes theorem, and the divergence theorem each relate differentiation on a region to integration on its boundary. This paper develops the language of smooth manifolds and differential forms in which these r…

Sidorenko's conjecture predicts that the homomorphism density of a bipartite graph H in a host graph G satisfies t_H(G) ≥ t_{K_2}(G)^|E(H)|. We introduce the conjecture through homomorphism densities, prove the finite-host inequality for P_3, C_4, and the complete bipartite grap…

In this paper we show that the chromatic function $P(G,t)$ is always a polynomial. We then ask a simple question: where does this polynomial equal zero? Tutte showed that two stretches of the real line, $(-\infty,0)$ and $(0,1)$, can never contain a root. Jackson showed in 1993 …

This paper primarily discusses the proof of the Mahler-Lech-Skolem through the uses of p-adic analysis in its connection to Strassman's Theorem and p-adic interpolation. As this theorem is so closely related to p-adic analysis, we make a brief mention of the core concepts in p-a…

Combinatorial game theory and coding theory appear to be completely separate fields at first glance. However, they share a profound connection through structures known as lexicographic codes, or lexicodes. First introduced by Conway and Sloane in 1986, lexicodes are generated by…

Hyperbolic geometry is a non-Euclidean geometry where geodesics play the role of straight lines. It is characterised by a constant negative curvature. The idea was conceived when mathematicians tried to understand the parallel postulate. This expository paper develops the founda…