Lectures related to Topological Data Science

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

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.

Algebraic Topology and Differential Geometry

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

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: second order

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

Manifolds and Tangent Space

Introduces manifolds, charts, atlases, tangent space, tensors, and the metric in curved spaces.

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

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

Momentum methods and nonlinear CG

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

Diophantine Approximation on Manifolds: Lower Bounds

Covers Diophantine approximation on manifolds with a focus on lower bounds.

Connections: Axiomatic Definition

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

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.

Truncated Conjugate Gradients for Trust-Region Subproblem

Explores truncated conjugate gradients for solving the trust-region subproblem in optimization on manifolds efficiently.

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.

Higher-order Interactions in Brain Networks

Explores higher-order interactions in brain networks using simplicial complexes and information theory, analyzing data from fMRI, financial time-series, and infectious diseases.

Differentiating Vector Fields: Definition

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

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.

Optimization on Manifolds Log in to Mediaspace to watch this video

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

Connections: motivation and definition Log in to Mediaspace to watch this video

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

General Manifolds and Topology Log in to Mediaspace to watch this video

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

Geodesics in Wasserstein Space Log in to Mediaspace to watch this video

Explores geodesics in the Wasserstein space, emphasizing constant speed geodesics and their properties.