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This paper investigates an important aspect of topological graph theory: methods for determining the genus of a graph. We discuss the classification of higher-order surfaces and then determine bounds on the genera of graphs embedded in orientable surfaces. After generalizing Euler's Formula to include graphs embedded on these surfaces, we derive upper and lower bounds for the genera of various families of simple graphs. We then examine some formulas for the genera of particular graphs.

The Poincare Lemma is a staple of rigorous multivariate calculus - however, proofs provided in early undergraduate education are often overly computational and are rarely illuminating. We provide a conceptual proof of the Lemma, making use of some tools from higher mathematics. The concepts here should be understandable to those familiar with multivariable calculus, linear algebra, and a minimal amount of group theory. Many of the ideas used in the proof are ubiquitous in mathematics, and the Lemma itself has applications in areas ranging from electrodynamics to calculus on manifolds.

We survey the field of combinatorial game theory. We discuss Zermelo's Theorem, a foundational result on which the theory of combinatorial game strategy is based. We then introduce the simple game of Nim and explain how it, through the theory of Nimbers, is critical to and underlies all of impartial combinatorial game theory.

Mathematicians have been interested in knot theory, or the study of knots, since the early nineteenth century. However, despite this interest, some basic questions remain unanswered; for example there is no effective way to definitively determine whether or not two knots are the same. In this paper, we will look at a powerful but frequently overlooked know invariant: the knot quandle. We will show that the knot quandle is a generalization of several more familiar invariants and that it is a complete invariant up to orientations. However, as we will see, determining whether two quandles are isomorphic is computationally intractable, which limits the utility of this otherwise powerful invariant.

This article presents a (very) brief overview of geometric problems involving tangent circles. In addition to defining the technique of inversion, we give two example problems with full solutions and suggest another challenge problem related to Pappus circles.

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