Conformal Blocks: Functions on Moduli Spaces

This lecture discusses conformal blocks and their role in the context of complex structures on surfaces. The instructor begins by explaining the concept of conformal blocks as functions defined on the deformation space or moduli space of complex structures. They elaborate on the relationship between complex structures and conformal classes, emphasizing the finite dimensionality of the space of complex structures with marked points. The discussion progresses to the challenges of defining canonical coordinates in higher genus surfaces and introduces Teichmüller theory as a framework for understanding these geometric complexities. The instructor highlights the significance of conformal blocks in conformal field theory (CFT) and their connections to algebraic geometry and quantization of moduli spaces. They also touch upon the construction of conformal blocks through L'Huvel theory and the importance of probabilistic methods in defining these mathematical objects. The lecture concludes with insights into the spectral resolution of operators associated with conformal blocks, emphasizing the interplay between analysis and probability in this context.