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Lecture
Modular Curves: Genus and Mapping Theorems
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Related lectures (30)
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.
Building surfaces from equilateral triangles
Explores the construction of Riemann surfaces from equilateral triangles and the dynamics of finite-type maps.
Meromorphic Functions & Differentials
Explores meromorphic functions, poles, residues, orders, divisors, and the Riemann-Roch theorem.
Modular curves: Riemann surfaces and transition maps
Covers modular curves as compact Riemann surfaces, explaining their topology, construction of holomorphic charts, and properties.
Open Mapping Theorem
Explains the Open Mapping Theorem for holomorphic maps between Riemann surfaces.
Harmonic Forms: Main Theorem
Explores harmonic forms on Riemann surfaces and the uniqueness of solutions to harmonic equations.
Proper Actions and Quotients
Covers proper actions of groups on Riemann surfaces and introduces algebraic curves via square roots.
Holomorphic Functions: Taylor Series Expansion
Covers the basic properties of holomorphic maps and Taylor series expansions in complex analysis.
Topology of Riemann Surfaces
Covers the topology of Riemann surfaces and the concept of triangulation using finitely many triangles.
Topology of Riemann Surfaces
Covers the topology of Riemann surfaces, focusing on orientation and orientability.
Harmonic Forms and Riemann Surfaces
Explores harmonic forms on Riemann surfaces, covering uniqueness of solutions and the Riemann bilinear identity.
Modular Forms: Dimension Formula
Explores modular forms, discussing pullback maps, meromorphic differentials, and the Riemann-Roch theorem.
Homology of Riemann Surfaces
Explores the homology of Riemann surfaces, including singular homology and the standard n-simplex.
Meromorphic Differentials and Modular Forms
Explores meromorphic differentials on Riemann surfaces and modular forms on congruence subgroups.
Modular Lambda Function: Properties and Applications
Explores the modular lambda function, its properties, and applications in modular forms.
Differential Forms Integration
Covers the integration of differential forms on smooth manifolds, including the concepts of closed and exact forms.
Canonical Divisors and Modular Forms
Covers canonical divisors on Riemann surfaces and properties of modular forms.
Functional Equation of Zeta
Covers the functional equation of zeta function and Jensen's formula in complex analysis.
Introduction to Quantum Chaos
Covers the introduction to Quantum Chaos, classical chaos, sensitivity to initial conditions, ergodicity, and Lyapunov exponents.
Riemann Surfaces: Complex Manifolds
Covers Riemann surfaces as complex manifolds of dimension 1, including transition maps and holomorphic functions.
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