Week One: Introduction to complex numbers, their geometry and algebra, working with complex numbers.
Week Two: The Mandelbrot set, Julia sets, a famous outstanding conjecture, history of complex numbers, sequences of complex numbers and convergence, complex functions.
Week Three: Complex differentiation and the Cauchy-Riemann equations.
Week Four: Conformal mappings, Möbius transformations and the Riemann mapping theorem.
Week Five: Complex integration, Cauchy-Goursat theorem, Cauchy integral formula, Liouville's Theorem, maximum principle, fundamental theorem of algebra.
Week Six: Power series representation of analytic functions, singularities, the Riemann zeta function, Riemann hypothesis, relation to prime numbers.