This lecture by José Ignacio Burgos Gil from ICMAT, Spain, explores Chern-Weil theory and the Hilbert Samuel theorem for singular metrics as part of the "Combinatorial Synergies in Geometry and Number Theory" workshop organized by the Instituto de Matemática Pura e Aplicada (IMPA). Discover how combinatorial methods intersect with algebraic geometry and number theory in this 46-minute presentation that bridges theoretical and computational aspects of mathematics. The workshop features mini-courses and presentations by international mathematicians focusing on topics like Berkovich spaces, tropical geometry, matroids, and polynomial equations, aiming to enhance combinatorial approaches in studying geometric and arithmetic structures.