Stefano Luzzatto

Chaos Theory

My field is popularly known as “Chaos Theory” and more technically as “Dynamical Systems” or “Smooth Ergodic Theory”. It starts from the basic observation, already well known to great classical mathematicians such as Laplace and Poincaré but brought to the general attention by Lorenz in the 1960’s, that in the world of differential equations and dynamical systems, the fact that “identical initial conditions have identical outcomes” does not imply that “almost identical initial conditions have similar outcomes”.

Indeed, we now understand that many systems exhibit very “sensitive dependence on initial conditions” meaning that extremely small changes in the initial conditions can lead very quickly to major changes in the outcomes. This has completely undermined a lot of strategies for understanding such systems which rely for example on numerical methods which are always approximations and thus susceptible to small errors in initial conditions and thus large errors in outcomes! A revolutionary approach was proposed in the 1960’s and 1970’s by Sinai who introduced the methods of statistical mechanics and the language of probability theory and suggested a “statistical” study of such systems. In the course of the following decades this has led to a huge amount of research and very deep results showing that many such “chaotic” systems are nevertheless well behaved and very stable from a statistical point of view.

Flipping Coins

As an example one can think of flipping a coin where the outcome of each flip is unpredictable but in the long run the average number of heads and tails tends to be the same, or even the weather which is unpredictable in the short term but where, for example, the average temperatures tend to be very stable. Most of my own research falls within the scope of a broad conjecture of Palis which can be informally stated as saying that “most systems are statistically well behaved”. It is a fascinating subject, both for its philosophical meaning and for the wide range of mathematical techniques involved, ranging from probability to topology to analysis and many other areas. Of particular interest are recent approaches which involve “rigorous computational methods” to obtain explicit and concrete numerical results.