Modular Curves: Genus and Mapping Theorems

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

Explores modular forms, discussing pullback maps, meromorphic differentials, and the Riemann-Roch theorem.

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

Explores meromorphic differentials on Riemann surfaces and modular forms on congruence subgroups.

Explores the modular lambda function, its properties, and applications in modular forms.

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

Covers canonical divisors on Riemann surfaces and properties of modular forms.

Covers the functional equation of zeta function and Jensen's formula in complex analysis.

Covers the introduction to Quantum Chaos, classical chaos, sensitivity to initial conditions, ergodicity, and Lyapunov exponents.

Covers Riemann surfaces as complex manifolds of dimension 1, including transition maps and holomorphic functions.