Explore local entropy theory and descriptive complexity in this advanced mathematics lecture from the Simons Semester on Dynamics series. Delve into the fundamental concepts of dynamical systems, focusing on topological and combinatorial descriptions of entropy occurrence. Learn how to apply local entropy theory to prove diverse mathematical results, including methods for identifying homoclinic points in algebraic actions and understanding the descriptive complexity of dynamical systems with completely positive entropy. Master key theoretical frameworks that bridge seemingly unrelated mathematical concepts through this nearly two-hour deep dive into advanced dynamical systems theory.