"Modular Forms and Computation"

Abstract: In this talk, we explore the theory and computational aspects of modular forms, which are complex functions characterized by an infinite number of symmetries. The talk begins by grounding the concept of mathematical symmetry in natural phenomena before transitioning to the rigid translational and inversional symmetries that define modular forms. The first half of the talk demonstrates the power of modular forms by applying them to classical problems in number theory.  The second half discusses methods for computing new, less-understood classes of these objects, specifically looking toward the frontier of noncongruence modular forms.