Concept

Smoothness

Related lectures (29)

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

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.

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.

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.

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

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.

Covers Eisenstein series properties and spectral decomposition of functions.

Covers Riemannian metrics, gradients, vector fields, and inner products on manifolds.

Explores obtaining n-th derivatives through a recursive process and the regularity of functions defined by power series.

Explores the transition from embedded to general manifolds, upgrading foundational concepts and discussing mathematical reasons for both approaches.

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

Covers the concept of bounding the Poisson bracket invariant on surfaces, exploring joint work with A. Logunov and S. Tanny.

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

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