Explore a lecture on counting closed geodesics and improving Weyl's law for predominant sets of metrics, delivered by Yaiza Canzani from UNC Chapel Hill at the first ZhengTong Chern-Weil Symposium in Mathematics. Delve into the typical behavior of two important quantities on compact manifolds with Riemannian metrics: the number of primitive closed geodesics of length smaller than T, and the error in the Weyl law for counting Laplace eigenvalues smaller than L. Discover the concept of predominance in the space of Riemannian metrics and learn about new stretched exponential upper bounds and logarithmic improvements that hold for a predominant set of metrics. Gain insights into joint work with J. Galkowski, which advances understanding beyond previous Baire-generic results from the 1970s and 1980s.