Explore the intriguing connection between quantum toric manifolds' non-commutative geometry and tame geometry in this 54-minute lecture by Ernesto Lupercio from CINVESTAV. Delve into explicit computations of Hodge numbers of LVM manifolds, which serve as a conjectural roadmap for further investigations into this relationship. The talk, presented at the University of Miami, is part of a collaborative research effort with L. Katzarkov, K.S. Lee, and L. Meersseman, offering insights into advanced topics in quantum toric geometry and motivic rings.