Explore complex toric quasifolds in this 55-minute lecture from the University of Miami, delivered by Fiametta Battaglia from Universita Degli Studi Firenze. Delve into the in-depth structure of these mathematical objects, discovering their shared key features with rational toric varieties, including the relationship between Betti numbers and polytope combinatorics. Follow the extension of constructions introduced in Elisa Prato's previous talk to nonsimple convex polytopes in both symplectic and complex settings, providing a comprehensive generalization of toric varieties to the nonrational case. Cover topics such as classical toric geometry, polytopes, complex and sympathetic quotients, algebra of n, combinatorics, proof of necessity, polytube, and complex nonsimple F vector.