Explore the intersection of noncommutative geometry and Hopf-cyclic cohomology in this 1-hour 6-minute lecture by Henri Moscovici from The Ohio State University. Delivered on September 28th, 2021, as part of the "Cyclic Cohomology at 40: achievements and future prospects" event at the Fields Institute, delve into the local index formula and its applications in noncommutative geometry. Gain insights into the mathematical foundations and recent developments in this specialized field, as Moscovici presents his research and expertise on the subject.