Fractals and Dynamics

I.- Schedule.

 Monday Tuesday Wed. Thursday Friday 9 – 10 Dynamics Dynamics Dynamics Dynamics Dynamics 10 – 11 Dynamics Dynamics Dynamics Dynamics Dynamics 11 – 12 Dynamics Dynamics Dynamics Dynamics Dynamics 12 – 1 Break Break Break Break Break 1 – 2 Fractals Fractals Fractals Fractals Posters 2 – 3 Fractals Fractals Fractals Fractals Posters 3 – 4 Fractals Fractals Fractals Fractals Posters

II.- Course on dynamics.

This course has mixed objectives. We will cover some prerequisites for the other activities of the course to take place, for example, complex numbers We will not engage into much details of these topics, rather learn to use them for the rest of the topics to make sense.

The main objective of the course is to explain how dynamics appeared historically: in the study of newton method for solving equations, but we will continue to study it development toward the works of Fatou and Julia. This will be a very engaging course: we will talk about math history, math problems, math ambient. The technicalities of this area are paramount, and we do not have the time to prepare for them, so we will do an approach based on examples, belief and history and supplement our intuition with further readings and exercises that appear in activities during the week.

The satellite objectives is to generate a sense of harmony within mathematics, a sense of awe at areas of math that are very rarely seen in highschool, and to engage the students into making their own explorations within this. It is expected that during the week the students ask questions to me, between themselves, or engage in finding their own paths.

III.- Fractals

In here we will study fractals, where do they appear, their properties and how do we study them. We will again take a very experimental approach due to the fact that many of the tool are very technical, but we will not try to avoid those technicalities but learn to handle them “intuitively” and to be able to answer questions and write solutions based on our explorations.

Based on what we do one day we will decide where to go the next day, what to explore, and which questions to try to answer. This will be a mixture of us discussing, experimenting and finding out, solving hand out problems related to our discussions and presenting solutions to the rest of the group.

IV.- Reading and Work

Everyone is going to get access to the book “The colours of Infinity”. Each student will pick the first day one of the articles in the book (they are independent) and study it through the week. The aim is that the student tries to understand the most he can of the topic at hand by making questions to me via email, or in person.

V.- Poster Presentation

During the week, based on the reading they selected and on the activities and problems thet are doing, eachs student will create a poster presentation in which public can engage with them. They should be able to explain their chosen topic (anything worked during the course or related to it that they might have found out during their own exploration) to the general lay public in a short presentation of 10 minutes approximately.

We will make a presentation with invited students, faculty, etc. of the math department. This should be taken seriously and the presentation should have quality. The point of the presentation is not to be extremely deep or technical, rather it is to give the student the opportunity to explain in their own way a part of what we are learning that they found particularly interesting.

VI.- Conclusion

We will ask every student to make a meditation on what they have learned, how they have grown during the course and feedback about this project and how it was handled.