Explore the geometric foundations and quantization theory of Landau-Ginzburg models in this 47-minute mathematical lecture. Delve into these natural objects that emerge in mirror symmetry studies, including how projective space mirror pairs correspond to LG models defined by Laurent polynomials. Examine the physics conjecture proposing equivalence between topological field theories of projective spaces and their corresponding LG models. Investigate the renowned LG/CY correspondence conjecture, which suggests that Gromov-Witten theory of Calabi-Yau hypersurfaces equals quantum singularity (FJRW) theory. Focus primarily on the geometric aspects of LG models while reviewing current quantization theory research and considering related open questions in this active area of mathematical physics.