Topological Data Science

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Related lectures (31)

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Retractions vector fields and tangent bundles: Tangent bundles

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

Riemannian connections

Explores Riemannian connections on manifolds, emphasizing smoothness and compatibility with the metric.

Orientation and degrees

Covers the concept of orientation and degrees in topology, focusing on assigning orientations to manifolds.

Optimality conditions: second order

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

Differentiating vector fields: Why do it?

Explores the importance of differentiating vector fields and the correct methodology to achieve it, emphasizing the significance of going beyond the first order.

Manifolds and Tangent Space

Introduces manifolds, charts, atlases, tangent space, tensors, and the metric in curved spaces.

Smooth sets and functions: Smooth functions, topology, and manifolds

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

Momentum methods and nonlinear CG

Explores gradient descent with memory, momentum methods, conjugate gradients, and nonlinear CG on manifolds.

Diophantine Approximation on Manifolds: Lower Bounds

Covers Diophantine approximation on manifolds with a focus on lower bounds.

Connections: Axiomatic Definition

Explores connections on manifolds, emphasizing the axiomatic definition and properties of derivatives in differentiating vector fields.