Problem sets and Presentation Topics

Here I will post Problem sets to learn the material covered in Lecture. In each Lecture there will be some time where you will be able to ask doubts about them. You will need to submit two homework assignments with 5 problems each of your choice. There will be a special emphasis in the clarity and presentation of your work.

Presentations:

Each student will do two presentations. This is one of the most important parts of the course, since you will be able to practice your presentation skills and the clarity in which you expose mathematics problems. At least one of the presentations will be a short lecture (30 minutes), like the one that you would do to your students in the future, explaining them a topic involving complex variables (which can range from High-School level to a more advanced Calculus course). If there is a topic, not in this list, that it is more of your interest, I am open to talk about it.
Remember that the focus in this course relies more in the exposition of the material rather than its difficulty. Once the course has started, whenever you decide for a topic, please let me know either via email or in person.

As soon as a presentation topic is taken it will look like this.

Here there are some suggestions for the future-like lecture:

1. Arithmetic of the Complex Numbers.
2. Geometric Interpretation of the Complex Numbers.
3. Euler's Identity.
4. Solution to polynomial equations.
5. Computing integrals of curves in the plane.
6. The Complex Exponential. Solution of some linear ODE's.
7. Complex Logarithm. Powers and roots of Complex Numbers.
8. Applications of the Residue Theorem: Computing integrals.
9. Applications of the Residue Theorem: Computing infinite sums.

Here there is a list of topics for the other presentation. These ones rely more in a deeper understanding and a more theoretical approach (however, the emphasis will still be more in the exposition).

1. The Riemann Sphere and the spherical metric.
2. Proving the Fundamental Theorem of Algebra.
3. The Problem of Analytic Continuation.
4. Cauchy's Theorem and Homotopy.
5. The Open Mapping Theorem and the Maximum Modulus Principle.
6. Harmonic Functions.
7. Applications of the Residue Theorem: Computing integrals.
8. Applications of the Residue Theorem: Computing infinite sums.
9. Application sof the Residue Theorem: The Argument Principle and counting zeros.
10. Applications of Complex Analysis to problems in Physics.
11. Normal Families and Montel's Theorem.
12. The Riemann Mapping Theorem.
13. Picard's Theorem.
14. Complex Dynamics: The Mandelbrot set.
15. Complex Dynamics: Newton's method to find zeros.
16. The Riemann Zeta Function.
17. The Hyperbolic metric.
18. Riemann surfaces and examples.