Explore enumerative geometry and its applications in algebraic geometry through a comprehensive lecture by Michail Savvas from the University of Texas at Austin. Delve into the world of geometric object counting, focusing on the relationship between geometric spaces like curves and more abstract concepts such as sheaves. Examine how sheaves generalize vector bundles and encode equations. Learn about moduli spaces and their role in parametrizing objects of interest, as well as the process of obtaining numerical invariants through integration of cohomology classes. Discover how this approach can be extended to cases where conventional assumptions do not hold. Investigate an important physics-motivated application: the construction of new generalized Donaldson-Thomas invariants for enumerating sheaves on Calabi-Yau threefolds. Gain insights from joint works with Young-Hoon Kiem and Jun Li in this one-hour Stony Brook Mathematics Colloquium presentation.