Lecture

Local Homeomorphisms and Coverings

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

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

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

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

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

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

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

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

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

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

Explores curves with Poritsky property, Birkhoff integrability, and Liouville nets in billiards.

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

Covers the normalization process of plane algebraic curves, focusing on irreducible polynomials and affine curves.

Explores mapping functions, surjections, injective and surjective functions, and bijective functions.

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

Explores the homology of Riemann surfaces, including singular homology and the standard n-simplex.

Delves into the rigidity of negatively curved manifolds and the interplay between curvature and symmetry.