Dynamical Systems

Dynamical Systems are an exciting branch of Mathematics. They use results and methods from many other branches and on the other hand find application in many branches of mathematics and other sciences. They are also recognized as important: two out of four Fields Medals (mathematical equivalent of Nobel prize) at the last International Mathematical Congress were awarded to researches in dynamics:  Artur Avila and Maryam Mirzakhani, by the way the first woman medalist in history. Four years earlier also two out of four medals were awarded for the results based on Dynamical Systems: Elon Lindenstrauss for applications of ergodic theory to number theory and  Stanislav Smirnov for applications to statistical physics.

The applications of Dynamical Systems start from those to “most pure” of mathematical disciplines - number theory. The relationship is a long lasting one: from Chebyshev’s results at the beginning of 20^th century through the works of Furstenberg,  Margulis, McMullen and Tao (three last are Fields medalist as well)  to the recent results of Lindenstrauss mentioned above.
Then come disciplines considered more practical Differential Equations and Probability Theory. The first is everyday bread and butter of every engineer,  the second of any statistician or financial specialist. The Differential Equations are inseparable from Dynamical Systems, for serious research they use dynamical methods, the notions of stability and all its variations are basic to Differential Equations.  Let us mention the famous Lorenz strange attractor which was a center of attention for the last 60 years.
Probability Theory uses many measure theoretic results which were first proven in Dynamical Systems. The most basic theorem of probability the Strong Law of Large Number is nothing else but Birkhoff Ergodic Theorem , also fundamental result of Dynamical Systems. Kolmogorov, the creator of mathematical Probability Theory was also one of the great figures of Dynamical Systems.
Let us go back to more abstract applications. One of the most famous mathematical objects is the Mandelbrot set, a beautiful fractal which is seen in innumerable images and represents a set studied in the dynamics of quadratic polynomials. Fractals have also practical applications, for example in material science. If the brakes of your car work properly it is also because the pads maker used methods based on fractal theory of surfaces. Another surprising application, every time you connect by your cell phone the connection is possible due to DS-CDMA algorithm based on a chaotic map of an interval, a toy model of every Dynamical Systems enthusiast.
Dynamical Systems methods are used also as theoretical tools in other sciences. In Statistical Physics the border between Mathematics and Physics became more ideological than real. Most recent results in Statistical Physics are actually results in Dynamical Systems, obtained both by mathematicians and  physicists, often working together. A new discipline, Mathematical Physics, was created and is blossoming. In Biology mathematical (Dynamical Systems) models successfully describe work of the heart and the propagation of diseases. In Chemistry the famous Belousov-Zhabotinsky reaction is an example of a periodic behavior and has numerous Dynamical Systems models.