Concept

Smooth scheme

Related lectures (31)

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

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

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

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

Presents a proof that smooth projective varieties with nef anti-canonical divisor have surjective Albanese morphism.

Explores étale motivic cohomology and the Hodge conjecture with a focus on counter-examples.

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

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

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

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

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

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

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

Covers the concept of local frames, their construction, and limitations.

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

Explores linear response in dynamical systems, emphasizing fractional derivatives and expanding unimodal maps.

Covers coordinate descent, a method for optimizing functions by updating one coordinate at a time.

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

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

Explores algebraic cycles, etale cohomology, and counterexamples to the Hodge conjecture.