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Abstract: Language is the most important evolutionary invention of the last few million years. How human language evolved from animal communication is a challenging question for evolutionary biology. In this paper we use mathematical models to analyze the major transitions in language evolution. We begin by discussing the evolution of coordinated associations between signals and objects in a population. We then analyze word-formation and its relationship to Shannon's noisy coding theorem. Finally, we model the population dynamics of words and the adaptive emergence of syntax.