Lectures related to Hermitian manifold

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

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.

Linear convergence with Polyak-Łojasiewicz: Mechanical proof

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

Geodesic Convexity: Basic Definitions

Introduces geodesic convexity on Riemannian manifolds and explores its properties.

Computing the Newton Step: Matrix-Based Approaches

Explores matrix-based approaches for computing the Newton step on a Riemannian manifold.

Riemannian Gradient Descent: Convergence Theorem and Line Search Method

Covers the convergence theorem of Riemannian Gradient Descent and the line search method.

Riemannian connections

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

Hermitian Forms: Definition and Properties

Explores the definition and properties of Hermitian forms in complex vector spaces.

Taylor expansions: second order

Explores Taylor expansions and retractions on Riemannian manifolds, emphasizing second-order approximations and covariant derivatives.

RTR practical aspects + tCG

Explores practical aspects of Riemannian trust-region optimization and introduces the truncated conjugate gradient method.

Geodesically Convex Optimization

Covers geodesically convex optimization on Riemannian manifolds, exploring convexity properties and minimization relationships.

Topology of Riemann Surfaces

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

Geodesic Convexity: Basic Facts and Definitions

Explores geodesic convexity, focusing on properties of convex functions on manifolds.

Trust Region Methods: Why, with an Example

Introduces trust region methods and presents an example of Max-Cut Burer-Monteiro rank 2 optimization.

Riemannian distance, geodesically convex sets

Covers the structure of Riemannian manifolds, geodesic convexity, and the Riemannian distance function.

Riemannian metrics and gradients: Computing gradients from extensions

Explores computing gradients on Riemannian manifolds through extensions and retractions, emphasizing orthogonal projectors and smooth extensions.

What is a smooth manifold? - Through defining functions

Explores smooth manifolds through defining functions and submanifolds, highlighting the importance of smoothness.

Riemannian metrics and gradients: Why and definition of Riemannian manifolds

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

Riemannian Hessians: Definition and Example

Covers the definition and computation of Riemannian Hessians on manifolds.

Newton's method on Riemannian manifolds

Covers Newton's method on Riemannian manifolds, focusing on second-order optimality conditions and quadratic convergence.