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Explore the final installment of a three-part mathematical seminar examining the mock modularity properties of Calabi-Yau threefolds and their applications to topological invariants. Focus on the computational aspects of polar terms in compact Calabi-Yau cases, where the solution of modular anomaly equations determines generating functions up to a finite number of coefficients. Learn how these polar terms can be calculated through wall-crossing techniques and direct integration of topological strings, leading to explicit modular and mock modular forms that encode rank 0 Donaldson-Thomas invariants. Discover how these results enable the generation of new Gopakumar-Vafa invariants for compact one-parameter threefolds, overcoming traditional limitations of direct integration methods. This advanced mathematical presentation builds upon previous discussions of modular forms, mock modular forms, and generalized Donaldson-Thomas invariants, providing concrete computational tools and applications in algebraic geometry and string theory.
