(E) We only hear about great mathematicians when they receive their Fields Metals or they solve a challenging problem. This week the four Field Metals winners, the equivalent of the Noble Prize in Mathematics, were announced on August 19 at the International Congress of Mathematicians in Hyderabad in India and included: Cédric Villani of the Henri Poincaré Institute in Paris, Stanislav Smirnov of the University of Geneva, Ngô Bao Châu of the University of Paris XI, and Elon Lindenstrauss of the Hebrew University of Jerusalem.

Professor Ngô proved in number theory the Fundamental Lemma, a building block of the Langlands Program that was initiated in 1967 by Professor Robert Langlands from Princeton University to unify certain aspects of algebra (Galois Representations) and analytics (Automorphic Forms). The Langlands Program led two other mathematicians to win as well the Fields Medals: Laurent Lafforgue and Vladimir Drinfel’d.

While most mathematicians have focused their researches in Theoretical Physics such as String Theory, Professor Villani has focused on applying probability theory to study the entropy in Fluid Dynamics and demonstrated not only that entropy increases, like when a gas escapes from a container, but it does it at different speeds, sometimes quickly or sometimes slowly. Professor Villani also worked on optimal transportation that defines how to ship, at the most effective cost, goods from a variety of producers to a variety of consumers.

Professor Smirnov has been rewarded for his research in a finite lattice model. A simple analogy to a two-dimension lattice is a chess board that stretches to the infinite and might have its lattices changed, as the lattices grow and change their shapes. Professor Smirnov’s research can be applied to both pure mathematics (complex dynamics) and applied mathematics (statistical physics).

Professor Lindenstrauss worked primarily in Ergodic Theory, a field of mathematics that analyses the behavior of a dynamical system when it is allowed to run over a long time. Ergodic Theory has applications as well as both in pure mathematics (number theory, differential geometry…) and applied mathematics (statistical physics).

While this year Field Medals achievements will probably only be remembered by the  mathematicians, that was not the case in 2006 when Grigori Perelman both refused his Fields Metals and his one million prize from the Clay Mathematics Institute to have proofed the Poincaré Conjecture, the first of the seven problems proposed by the Clay Institute to be solved. The Poincaré Conjecture basically states that “the sphere is the only three-dimensional closed space lacking holes” or better stated in 1904 by Henri Poincare himself and in his own words “la sphere est le seul espace compact simplement connexe de dimension trois”. While the Poincaré conjecture is easy to consider in two-dimensional spaces, it is much harder to conceive for higher dimensions: Stephen Smale solved it for five dimensions and Michael Freedman for six dimensions. And, both of them were rewarded by the Field Medals. Dr. Perelman’s proof, very simply said, leverages the Ricci Flow from Richard Hamilton to re-shape, cut and modify a surface that has three dimensions to conclude that it is a sphere.

By solving hard problems Mathematicians are developing critical theories leading to new scientific work and technologies from fast algorithms for searching DNA sequences in biology, to models to analyze the behaviors of the returns of an asset in finance.

Mathematics is still essential for the progress of many sciences and technologies.

For instance, pioneered by Benoit Mandelbrot in the 1980s, Fractal Geometry establishes the repetition of geometric patterns at different scales in nature. Fractal Geometry has been applied to many sciences and technologies in particular biology, meteorology, seismology, finance, and computer networking! The fundamental assumption of Fractal Geometry is that fractals kept the same shape or properties over different scales. Its mathematics can be leveraged to study any type of patterns in space or time that remain the same even as the scale of the observed changes.

Over the last few years, mathematics has been key to better understand our Universe. Finally, theoretical physicists have reconciled the antagonisms between quantum mechanics and Einstein’s relativity by proposing initially the String Theory which has become now the M-Theory. Furthermore, some physicists have leveraged String Theory to propose that the origin of the Big Bang came from the collision of two branes (a concept issued from String Theory).

And, probably no other mathematicians have pushed recently the frontiers of mathematics than cosmologist expert in Parallel Universes, Professor Max Tegmark from MIT. Professor Tegmark argued that “our physical reality is a mathematical structure and that our universe is not just described by mathematics – it is mathematics”. According to Professor Tegmark’s Mathematical Universe Hypothesis (MUH), “we don’t invent mathematical structures – we discover them, and invent only the notation for describing them”.

Whatever is the problem being solved or the application of the theory, mathematics is still essential for the progress of many sciences and technologies and not only physics but also engineering, computer science, biology, finance and economy, why you can agree or disagree with Professor Tegmark if the Universe is Maths or not someone has still to establish the equations to explain “why does water boil when it is warming up?”.

References

• Terence Tao’s Blog article on Lindenstrauss, Ngo, Smirnov, Villani
• The Millennium Prize Problems stated by the Clay Mathematics Institute
• The Math Book, From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics, Clifford A. Pickover, Sterling Publisher

Note: The picture above is a sculpture from Giulio Monteverde, “Columbus as a Boy”.

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