Dynamical Systems is the part of mathematics which studies the changes that occur as a system evolves over time. Some systems are easy to predict whereas others are not. Such difficult to predict systems are known as "chaotic systems" (for example, the stock market or the weather). It is a surprizing fact that very simple transformations can give rise to very complex (and beautiful !) behavior. In this mini-course we will start by describing the dynamics under iteration of a simple family of maps (real quadratic polynomials) and we will work our way through to the study of the dynamics of quadratic complex polynomials. We will tour the parameter space of complex quadratic polynomials, known as the Mandelbrot set, and will describe the Fatou and Julia sets for some parameters.

There are many beautiful identities involving positive integers. For example, Pythagoras knew 3² + 4² = 5² while Plato knew 3³ + 4³ + 5³ = 6³. Euler discovered 59⁴ + 158⁴ = 133⁴ + 134⁴, and even a famous story involving G. H. Hardy and Srinivasa Ramanujan involves 1³ + 12³ = 9³ + 10³. But how does one find such identities? Around the third century, the Greek mathematician Diophantus of Alexandria introduced a systematic study of integer solutions to polynomial equations. In this session, we'll focus on various types of so-called Diophantine Equations, discussing such topics as the Postage Stamp Problem, Pythagorean Triples, Pell's Equations, Elliptic Curves, the ABC Conjecture and Fermat's Last Theorem.