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Concept

Manifold

Related lectures (31)

Differential Forms on Manifolds

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

Topology of Riemann Surfaces

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

Differential Forms Integration

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

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.

Optimality Conditions: First Order MOOC: Introduction to optimization on smooth manifolds: first order methods

Covers optimality conditions in optimization on manifolds, focusing on global and local minimum points.

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.

Local Homeomorphisms and Coverings

Covers the concepts of local homeomorphisms and coverings in manifolds, emphasizing the conditions under which a map is considered a local homeomorphism or a covering.

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.

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.

Algebraic Topology and Differential Geometry

Explores algebraic topology and differential geometry in understanding robot dynamics and global behavior.

Optimality conditions: second order

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

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

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

Fundamental Groups

Explores fundamental groups, homotopy classes, and coverings in connected manifolds.

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.

Connections: Axiomatic Definition

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

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 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.

Group Theory in Physics

Covers the fundamentals of group theory in physics, focusing on symmetries and transformations leaving physical equations unchanged.

Momentum methods and nonlinear CG

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