Lectures related to Bird's-eye view and aims

Differential Forms on Manifolds

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

Algebraic Topology and Differential Geometry

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

Riemannian connections: What they are and why we care

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

Riemannian connections: Proof sketch

Presents the fundamental theorem of Riemannian geometry and demonstrates the uniqueness of the Riemannian connection.

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.

Topology of Riemann Surfaces

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

Riemannian Hessians: Connections and Symmetry

Covers connections on manifolds, symmetric connections, Lie brackets, and compatibility with the metric in Riemannian geometry.

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.

Differential Forms Integration

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

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.

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.

Differentiating Vector Fields: Definition

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

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.

Rigidity in Negative Curvature

Delves into the rigidity of negatively curved manifolds and the interplay between curvature and symmetry.

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

Explains Riemannian submanifolds, metrics, and gradients computation on manifolds.

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.

Riemannian Hessians: Definition and Example

Covers the definition and computation of Riemannian Hessians on manifolds.

Geodesics and Parallel Transport

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

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.

Connections: Axiomatic Definition

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