Low Dimensional Topology and Circle-valued Morse Functions

This course covers the following learning outcomes and goals: understanding the representations of fundamental groups with values in Homeo_+(S^1), exploring the verification of left-orderability of 3-manifold groups in various scenarios, such as toroidal integer homology spheres, surgeries on knots, and cyclic branched covers of hyperbolic links. The course also delves into the relationship between these representations and the L-space conjecture, as well as the existence of taut foliations in connection to the conjecture. The course teaches the individual skills of analyzing foliations and flows on 3-manifolds, interpreting Homeo_+(S^1) representations, and applying these concepts to verify left-orderability of 3-manifold groups. The teaching method of the course involves lectures, discussions, and joint work presentations based on research conducted with other scholars. The intended audience for this course includes researchers, scholars, and students interested in low-dimensional topology, circle-valued Morse functions, and the L-space conjecture in the context of 3-manifolds.