Introduction: Fields Medal Symposium 2013

Welcome to the 2013 version of the Fields Medal Symposium blog! I am Hector Pasten and I will be writing for you about the 2013 Fields Medal Symposium during the next couple of weeks.
This year the symposium will take place at the Fields Institute, Toronto, from September 30 to October 3, in honor of Professor Elon Lindenstrauss who was awarded the Fields Medal in the International Congress of Mathematicians (ICM) held in India, 2010.

To get started, let’s have a quick look at some of main topics related to this year’s event. This information will be expanded in forthcoming posts.

The Fields Medal.

This is the top honor that a young researcher in mathematics can aim for. If you are under 40 years old and have produced revolutionary advances in mathematics then you may qualify for it.
Originally, it started under the name of “International Medal for Outstanding Discoveries in Mathematics” and was first awarded at the ICM held in Norway, 1936, to Lars Ahlfors and Jesse Douglas. It later became known as Fields Medal in memory of John Charles Fields, a Canadian mathematician who established this award (who even designed the medal itself!).

Elon Lindenstrauss.

Professor Lindenstrauss is a mathematician working in Ergodic Theory, Dynamical systems and applications to Number Theory. He was born in 1970, Jerusalem, Israel, and obtained his PhD at the Hebrew University in 1999. His thesis has the title “Entropy properties of dynamical systems”.
Because of his deep contributions to mathematics, he was awarded the Fields Medal at the ICM held in India, 2010, along with Ngo Bao Chau, Stanislav Smirnov and Cedric Villani. More precisely, according to the web-page of the ICM 2010, Professor Lindenstrauss was awarded the 2010 Fields Medal for his results on measure rigidity in ergodic theory, and their applications to number theory.

Ergodic Theory.

If somebody on the street asks you “what is Ergodic Theory about?” here is a short answer which is not too far from the truth: is the theory that explains why if you put 10 litres of white paint and 1 litre of black paint in a container and you shake it, then you can expect to get 11 litres of grey paint.
A main topic in Ergodic theory is understanding (using invariant measures, group actions, test functions, equidistributed sequences, etc.) when is it true that the space average in a dynamical process equals the time average.
In our example of white and black paints, the “space average” is the initial information of having white and black paint in a proportion 10:1, while “time average” corresponds to the final color (grey) that you get after the dynamical process of shaking the container for a while.

Number Theory.

Again, if the same individual on the street asks you “what is Number Theory?”, here is a short answer that might make him happy: it is the theory that investigates the properties of whole numbers and their close relatives (rational numbers, algebraic numbers, polynomials over finite fields, holomorphic functions, modular forms, etc.).
Of course, the trick is what we mean with “close relatives” but let’s not discuss this point here. Instead, let us mention that several areas in Number Theory have been greatly benefited by the developments in Ergodic Theory. This is the case for instance with quadratic forms, Diophantine approximation, Modular forms and special values of L-functions.

Connections between Ergodic Theory and Number Theory.

Here is a simple statement that can give an idea of the connection between Number Theory and Ergodic Theory: if x is an irrational real number, then the sequence x, 2x, 3x, … contains terms that are arbitrarily close to integers. If you think in terms of Ergodic Theory, then you would like to prove this in a dynamical way: the fractional parts of these numbers are uniformly distributed in the interval [0,1] with respect to the Lebesgue measure (in particular, there are fractional parts arbitrarily close to 0, hence the result). Of course, there are other ways to approach this simple problem, but in some cases Ergodic Theory really gives the key for opening doors that remained closed for a long time. For instance, this is the case with the Oppenheim conjecture about quadratic forms, which we hopefully will discuss in a later post.

Stay tuned, and let’s look forward to the 2013 Fields Medal Symposium!

About hpasten

I am mainly interested in Number Theory.
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