Explore the intricate relationship between optimal transport theory and the construction of complete Calabi-Yau metrics on log Calabi-Yau varieties in this differential geometry and physics seminar lecture. Delve into how the geometric challenge of constructing complete Calabi-Yau metrics naturally leads to boundary regularity problems in optimal transport theory. Discover how geometric insights can advance understanding of these transport problems and learn about cutting-edge research connecting two fundamental areas of mathematics. Examine collaborative work that bridges differential geometry, algebraic geometry, and optimal transport, revealing deep connections between seemingly disparate mathematical fields. Gain insight into current research directions in Calabi-Yau geometry and its applications to theoretical physics, particularly in understanding the geometric structures that arise in string theory and mirror symmetry.