Lectures related to Compact Manifolds Classification

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

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

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

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.

Harmonic Forms: Main Theorem

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

Meromorphic Functions & Differentials

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

Differential Forms and Invariant Measures

Covers differential forms, invariant measures, and integration on manifolds with examples and illustrations.

Topology of Riemann Surfaces

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

Proper Actions and Quotients

Covers proper actions of groups on Riemann surfaces and introduces algebraic curves via square roots.

Fundamental Groups

Explores fundamental groups, homotopy classes, and coverings in connected manifolds.

Connections: motivation and definition

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

Projective Varieties: An Algebraic Study

Covers the study of projective varieties and their relation to compact manifolds.

Complex Manifolds: GAGA Principle

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

Integration Theory: Berkovich Spaces

Explores integration theory over real numbers and Berkovich spaces, revealing intriguing asymmetries and unsolved conjectures.

Automorphisms of Projective Varieties

Explores automorphisms of projective varieties, discussing isomorphism between complements and key observations on dimensions and examples.

Smooth Projective Varieties

Covers regular and smooth projective varieties, hypersurfaces, and algebraic dimensions.

Albanese Morphism: Kodaira Dimension Zero

Explores the Albanese morphism of varieties with Kodaira dimension zero in positive characteristic.

Dynamics on Homogeneous Spaces and Interactions with Number Theory

Delves into Oppenheim's conjecture on quadratic forms and their connection to number theory.

Varieties with nef anti-canonical: Surjective Albanese

Presents a proof that smooth projective varieties with nef anti-canonical divisor have surjective Albanese morphism.

Regularity and Geometric Meaning

Explores regularity in algebraic geometry and the geometric implications of the Jacobian criterion.