Explore the fundamental question of geometric representation in semantic modeling through this hour-long lecture delivered at Johns Hopkins University's Center for Language and Speech Processing. Examine the theoretical foundations and practical implications of using geometric approaches to model meaning in computational linguistics and natural language processing. Delve into the debate surrounding the optimal level of geometric structure needed in semantic representations, considering both the benefits and limitations of spatial metaphors for capturing linguistic meaning. Analyze various approaches to semantic modeling and their geometric properties, while evaluating how different levels of geometric constraint affect model performance and interpretability in language understanding tasks.