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Recent decades have shown a trend of mathematicians using various methods of algebra, geometry, dynamical systems, and even representation theory to study problems involving aggregation methods. One area that has sparked interest is Voting Theory, the study of formally defined voting systems. We will begin with Arrow's Impossibility Theorem, which demonstrates that no voting system can possibly meet a certain set of reasonable criteria. We will look at the most popular voting systems in practice and under study including Borda Count, Condorcet, Plurality and Single Transferable Voting. We will demonstrate faults of these systems and paradox's that may occur from their use. We will then look in detail into the methods of Donald Saari PhD. who studies voting structures as naturally arising vector spaces, and hence admitting geometries that can be analyzed. Saari's contributions have helped to classify a large class of election paradoxes and suggest remedies to these paradox's by exploiting symmetries of the associated vector spaces.