Lectures related to Smooth structure

Covers the integration of differential forms on smooth manifolds, including the concepts of closed and exact forms.

Covers retractions, tangent bundles, and embedded submanifolds on manifolds with proofs and examples.

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

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

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

Explores smooth functions, n-dimensional charts, compatibility, and atlases for manifolds.

Covers the approximation of functions in Sobolev spaces using smooth functions.

Explores smooth maps, differentials, composition properties, linearity, and extensions on manifolds.

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

Explores smooth functions on manifolds, emphasizing continuity and atlas topologies.

Explores upgrading foundations from embedded to general manifolds in optimization, discussing smooth sets and tangent vectors.

Covers smooth maps on manifolds, defining functions, tangent spaces, and differentials.

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

Explores smooth manifolds through diffeomorphisms and embedded submanifolds in a linear space.

Covers the Laplace equation, decomposition of linear problems, and solutions through separation of variables.

Discusses advanced optimization techniques, focusing on faster and projected gradient descent methods in machine learning.

Introduces retractions and vector fields on manifolds, providing examples and discussing smoothness and extension properties.

Explores Taylor expansions and retractions on Riemannian manifolds, emphasizing second-order approximations and covariant derivatives.

Discusses approximation by smooth functions and the convergence of function sequences in normed vector spaces.

Covers smooth sets and functions on manifolds, with a focus on constructing an atlas for a circle.