Lectures related to Riemannian connections: What they are and why we care

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

Hessians, Symmetry and Examples: Sphere, Stiefel

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

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.

Connections: motivation and definition

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

Symmetry Property: Riemannian Connection in Geometry

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

Optimization on Manifolds

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

Riemannian connections: Proof sketch

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

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

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

All things Riemannian: metrics, (sub)manifolds and gradients

Covers the definition of retraction, open submanifolds, local defining functions, tangent spaces, and Riemannian metrics.

Hands on with Manopt: Optimization on Manifolds

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

Differential Forms on Manifolds

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

Connections: Axiomatic Definition

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

From embedded to general manifolds: Why?

Explores upgrading foundations from embedded to general manifolds in optimization, discussing smooth sets and tangent vectors.

Rigidity in Negative Curvature

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

Gradients on Riemannian submanifolds, local frames

Discusses gradients on Riemannian submanifolds and the construction of local frames.

Riemannian metrics and gradients: Examples and Riemannian submanifolds

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

Retractions vector fields and tangent bundles: Tangent bundles

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

Comparing Tangent Vectors: Parallel Transport

Explores the definition, existence, and uniqueness of parallel transport of tangent vectors on manifolds.