Lectures related to Riemann Surfaces: Complex Manifolds

Topology of Riemann Surfaces

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

Harmonic Forms: Main Theorem

Explores harmonic forms on Riemann surfaces and the uniqueness of solutions to harmonic equations.

Harmonic Forms and Riemann Surfaces

Explores harmonic forms on Riemann surfaces, covering uniqueness of solutions and the Riemann bilinear identity.

Meromorphic Functions & Differentials

Explores meromorphic functions, poles, residues, orders, divisors, and the Riemann-Roch theorem.

General Manifolds and Topology

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

Modular curves: Riemann surfaces and transition maps

Covers modular curves as compact Riemann surfaces, explaining their topology, construction of holomorphic charts, and properties.

Topology of Riemann Surfaces

Covers the topology of Riemann surfaces and the concept of triangulation using finitely many triangles.

Holomorphic Functions: Taylor Series Expansion

Covers the basic properties of holomorphic maps and Taylor series expansions in complex analysis.

Manifolds: Charts and Compatibility

Covers manifolds, charts, compatibility, and submanifolds with smooth analytic equations.

Differential Forms on Manifolds

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

Complex Manifolds: GAGA Principle

Covers the GAGA principle, stating that any morphism on projective varieties is constant.

Open Mapping Theorem

Explains the Open Mapping Theorem for holomorphic maps between Riemann surfaces.

Building surfaces from equilateral triangles

Explores the construction of Riemann surfaces from equilateral triangles and the dynamics of finite-type maps.

Conformal Blocks: Functions on Moduli Spaces

Covers conformal blocks and their significance in complex structures and moduli spaces.

Differential Forms Integration

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

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

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

Orientation and degrees

Covers the concept of orientation and degrees in topology, focusing on assigning orientations to manifolds.

Local Homeomorphisms and Coverings

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.

Analytic Manifolds and Berkovich Spaces

Introduces analytic manifolds, Berkovich spaces, and multiplicative semi-norms in K-algebras.

Curl and Exact Sequences

Covers the concept of curl in vector calculus and De Rham cohomology.