Lectures related to Smoothness

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

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

Introduces differentiating vector fields along curves on manifolds with connections and the unique operator satisfying specific properties.

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

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

Covers the Quadratic Sieve method for integer factorization, emphasizing the importance of choosing the right parameters for efficient factorization.

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

Explores non-analytic smooth functions, their properties, and applications in differential geometry and partitioning unity.

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

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 smooth functions on manifolds, emphasizing continuity and atlas topologies.

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

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

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

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

Explores bijective functions, coordinate transformations, and graphical representations in two dimensions.

Explores necessary and sufficient optimality conditions for local minima on manifolds, focusing on second-order critical points.

Covers remarks and demonstrations on function continuity and differentiability, focusing on class C functions.

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