Math 740: Foundations of Differential Geometry is a graduate-level course focusing on connections and curvature on manifolds, with a deeper exploration in the context of Riemannian and Lorentzian metrics. The course emphasizes the study of geodesics and their relationship with curvature. Students will learn to compute tensors, such as metrics and curvature, both in coordinates and coordinate-free forms. Learning outcomes include a detailed understanding of the exponential map, criteria for completeness, conjugate points, and Jacobi fields. Students will also gain an understanding of curvature's multiple interpretations, such as its role as an obstruction to local parallel framing and its influence on the spreading of geodesics through the Jacobi equation. The course includes a real-world application: the precession of Mercury's perihelion, which can be explained using Lorentzian differential geometry to derive and solve the relativistic equation for the orbit, a geodesic in the Schwarzschild model of spacetime. The course format includes two 75-minute lectures per week, semi-weekly graded problem sets, and a written final exam. Recommended resources include "Semi-Riemannian Geometry, with Applications to Relativity" by B. O'Neill and "Introduction to Smooth Manifolds" by J.M. Lee (2nd edition).Math 740: Foundations of Differential Geometry is a graduate-level course focusing on connections and curvature on manifolds, with a deeper exploration in the context of Riemannian and Lorentzian metrics. The course emphasizes the study of geodesics and their relationship with curvature. Students will learn to compute tensors, such as metrics and curvature, both in coordinates and coordinate-free forms. Learning outcomes include a detailed understanding of the exponential map, criteria for completeness, conjugate points, and Jacobi fields. Students will also gain an understanding of curvature's multiple interpretations, such as its role as an obstruction to local parallel framing and its influence on the spreading of geodesics through the Jacobi equation. The course includes a real-world application: the precession of Mercury's perihelion, which can be explained using Lorentzian differential geometry to derive and solve the relativistic equation for the orbit, a geodesic in the Schwarzschild model of spacetime. The course format includes two 75-minute lectures per week, semi-weekly graded problem sets, and a written final exam. Recommended resources include "Semi-Riemannian Geometry, with Applications to Relativity" by B. O'Neill and "Introduction to Smooth Manifolds" by J.M. Lee (2nd edition).