Explore the intricacies of virtual invariants in moduli spaces of sheaves on surfaces and Vafa-Witten invariants in this comprehensive lecture by Lothar GOETTSCHE. Delve into topics such as logical invariants, mathematical results, singular spaces, virtual order, and general versions. Examine the concept of virtual localization and integration of fixed points. Gain insights into modularity, section numbers, and line runners. Enhance your understanding of advanced mathematical concepts in this in-depth seminar from ICTP Mathematics.