Diophantine Approximation on Manifolds: Lower Bounds

Covers the topology of Riemann surfaces, focusing on orientation and orientability.

Explores the definition of connections for smooth vector fields on manifolds.

Introduces differential forms on manifolds, covering tangent bundles and intersection pairings.

Covers manifolds, topology, smooth maps, and tangent vectors in detail.

Covers optimization on manifolds, focusing on smooth manifolds and functions, and the process of gradient descent.