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Explore the foundational concepts of modular forms, mock modular forms, Jacobi forms, and indefinite theta series in this advanced mathematics lecture from the M-Seminar at Kansas State University. Delve into the mathematical framework underlying generalized Donaldson-Thomas (DT) invariants associated with Calabi-Yau threefolds, focusing on their modular properties and computational applications. Learn how these modular properties enable the calculation of rank 0 DT invariants and understand their broader implications for other topological invariants including Vafa-Witten and Gopakumar-Vafa invariants. Begin with essential background on various types of modular forms before transitioning to the specific context of generalized DT invariants in algebraic geometry. This lecture serves as the first installment of a comprehensive three-part series that will ultimately demonstrate how modular anomaly equations can be solved for both compact and non-compact Calabi-Yau threefolds, leading to explicit computations of topological invariants through wall-crossing techniques and topological string integration methods.
