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Possible Topics (Note: this is by no means an exhaustive list; feel free to approach me about speaking on another topic): Thue-Siegel-Roth Theorem, Subspace Theorem, Linear Forms In Logarithms, Class Number 1 Problem, Irrationality and Q-linear Independence of Zeta Values, Khintchine Theorem, Oppenheim Conjecture, Siegel's Theorem on Integral Points, Faltings' proof of Mordell-Lang

Some useful sources (again, by no means exhaustive): 

Baker, Transcendental Number Theory

Bombieri, Gubler, Heights in Diophantine Geometry

Cassels, An Introduction to Diophantine Approximation

Schmidt, Diophantine Approximation