Lectures related to Bird's-eye view and aims

Riemannian connections

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

Differential Forms on Manifolds

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

Connections: motivation and definition

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

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.

Optimization on Manifolds

Covers optimization on manifolds, focusing on smooth manifolds and functions, and the process of gradient descent.

Riemannian Hessians: Connections and Symmetry

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

Manopt: Optimization Toolbox for Manifolds

Introduces Manopt, a toolbox for optimization on manifolds, focusing on solving optimization problems on smooth manifolds using the Matlab version.

Optimization on Manifolds: Context and Applications

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.

Symmetry Property: Riemannian Connection in Geometry

Explores symmetries, Riemannian connection, vector fields, and Lie bracket in geometry.

General Manifolds and Topology

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

Riemannian metrics and gradients: Why and definition of Riemannian manifolds

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

Linear Lie Groups: Definitions and Theorems

Discusses linear Lie groups, their definitions, properties, and the relationship between integral curves and vector fields.

Retractions vector fields and tangent bundles: Tangent bundles

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

Non-analytic Smooth Functions

Explores non-analytic smooth functions, their properties, and applications in differential geometry and partitioning unity.

Hessians, Symmetry and Examples: Sphere, Stiefel

Covers Hessians, symmetry, and examples related to vector fields, functions, and manifolds.