Lectures related to Contact geometry

Forcing of Topological Entropy

Delves into topological entropy in compact manifolds and Reeb flows, emphasizing the forcing of entropy via cylindrical contact homology.

Symplectic Rigidity: Forms of Symplectic Rigidity

Explores symplectic rigidity, including rigidity, flexibility, and dynamical rigidity, with a focus on symplectic manifolds and Lagrangian submanifolds.

Smooth maps and differentials: Smooth maps

Explores smooth maps between submanifolds, discussing smoothness criteria and composition preservation.

Submanifolds: Locally Deformable into Linear Patches

Explores how submanifolds can be deformed into linear patches using diffeomorphisms and the inverse function theorem.

Smooth maps on manifolds and differentials

Covers smooth maps on manifolds, defining functions, tangent spaces, and differentials.

Differentiating Vector Fields: How Not to Do It

Discusses the challenges in differentiating vector fields on submanifolds and the importance of choosing the right method.

Retractions vector fields and tangent bundles: Tangent bundles

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

Fixed-Rank Manifold: Tangent Spaces and Local Defining Functions

Explores fixed-rank matrices as an embedded submanifold, focusing on constructing local defining functions and efficiently computing tangent vectors.

Differential Forms Integration

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

Embedded Submanifolds: Stiefel Manifold

Covers embedded submanifolds, Stiefel manifold, tangent spaces, and differential ranks.

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

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

Smooth Manifolds: Diffeomorphisms

Explores smooth manifolds through diffeomorphisms and embedded submanifolds in a linear space.

Smooth Manifolds: Setup

Introduces smooth manifolds, emphasizing the importance of submanifolds of linear spaces.

Periodic Orbits of Hamiltonian Systems

Covers the theory of periodic orbits of Hamiltonian systems and the Conley-Zehnder index.

Volume Growth of Positive Contactomorphisms

Covers the growth of positive contactomorphisms in Legendrian spheres and symplectic forms.

From embedded to general manifolds: upgrading our foundations

Explores the transition from embedded to general manifolds, upgrading foundational concepts and discussing mathematical reasons for both approaches.

What is a smooth manifold? - Through defining functions

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

Tangent spaces: Linearization of Embedded Submanifolds

Explores tangent spaces as free movement directions on submanifolds, offering a geometrically satisfying linearization notion.

Riemannian metrics and gradients: Computing gradients from extensions

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

Grassmann manifold and Retractions

Covers the Grassmann manifold and retractions on submanifolds.