Lectures related to Taylor expansions: second order

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

Differentiating Vector Fields: Definition

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

Riemannian metrics and gradients: Why and definition of Riemannian manifolds MOOC: Introduction to optimization on smooth manifolds: first order methods

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

Gradient Descent MOOC: Introduction to optimization on smooth manifolds: first order methods

Explores gradient descent methods for optimizing functions on manifolds, emphasizing small gradient guarantees and global convergence.

Differentiating vector fields along curves: Examples and finite differences

Covers the differentiation of vector fields along curves with examples and finite differences.

Differential Forms Integration

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

Riemannian connections: What they are and why we care

Covers Riemannian connections, emphasizing their properties and significance in geometry.

Dynamics of Steady Euler Flows: New Results

Explores the dynamics of steady Euler flows on Riemannian manifolds, covering ideal fluids, Euler equations, Eulerisable flows, and obstructions to exhibiting plugs.

Retractions vector fields and tangent bundles: Tangent bundles MOOC: Introduction to optimization on smooth manifolds: first order methods

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

Optimization on Manifolds: Context and Applications MOOC: Introduction to optimization on smooth manifolds: first order methods

Introduces optimization on manifolds, covering classical and modern techniques in the field.

Hands on with Manopt: Optimization on Manifolds MOOC: Introduction to optimization on smooth manifolds: first order methods

Introduces Manopt, a toolbox for optimization on smooth manifolds with a Riemannian structure, covering cost functions, different types of manifolds, and optimization principles.

Geodesics and Parallel Transport

Explores geodesics, parallel transport, and the Riemann tensor on two-dimensional manifolds, emphasizing fundamental concepts in differential geometry.

Riemannian metrics and gradients: Examples and Riemannian submanifolds MOOC: Introduction to optimization on smooth manifolds: first order methods

Explores Riemannian metrics on manifolds and the concept of Riemannian submanifolds in Euclidean spaces.

Smooth maps and differentials: Differentials MOOC: Introduction to optimization on smooth manifolds: first order methods

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

Riemannian Hessians: Definition and Example

Covers the definition and computation of Riemannian Hessians on manifolds.

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.

Linear convergence with Polyak-Łojasiewicz: Mechanical proof

Explores linear convergence with the Polyak-Łojasiewicz condition on a Riemannian manifold.

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

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

Geodesic Convexity: Basic Definitions

Introduces geodesic convexity on Riemannian manifolds and explores its properties.

Connections: motivation and definition Log in to Mediaspace to watch this video

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