The 2010 Chern Medal to Louis Niremberg

August 19, 2010

The inaugural award of the Chern Medal has gone to Louis Niremberg of the Courant institute in New York for his role in the formulation of the modern theory of non-liner elliptic partial differential equations. The new mathematics prize in honour of Shiing-Shen Chern comes with a cheque for $250,000.

Niremberg is a Canadian born mathematician who is renowned for his fundamental work on partial differential equations and their applications in complex analysis.

The 2010 Gauss Prize to Yves Meyer

August 19, 2010

The Gauss Prize is for mathematics applied to modern technologies and was given for the first at the IMU conference in 2006. This years prize has been awarded to Yves Meyer for his work on wavelets.

Wavelets are sometimes called brief oscillations because they consist of a short wavelike curve that is zero outside its finite bounds. Wavelet analysis is like a cross between Fourier analysis and B-splines which combines the advantages of both. It can be applied to efficiently encoding sound, images and other signals and its impact on digital technologies of recent years has been enormous.

The 2010 Nevanlinna Prize to Daniel Spielman

August 19, 2010

Daniel Spielman is a computer scientist and mathematician at Yale University. he has been awarded the Nevanlinna Prize for his contribution to information technology including smoothed analysis, a new way of measuring the complexity of an algorithm.

A Fields Medal for Stanislav Smirnov

August 19, 2010

Stanislav Smirnov is a Russian mathematician working at the university of Geneva  on complex analysis, dynamical systems and probability theory. He proved Cardy’s formula for critical site percolation on the triangular lattice and deduced conformal invariance.

A Fields Medal for Elon Lindenstrauss

August 19, 2010

Elon Lindenstrauss was born in Israel in 1970. He has been awarded the Fields medal for his work on ergodic theory (the study of measure preserving transformations) and its applications to number theory. Using his methods he has made significant progress on the Littlewood conjecture which remains an open problem in Diophantine approximation.

Lindeanstrauss also proved the arithmetic quantum unique ergodicity conjecture of Rudnick and Sarnak which relates to modular forms. Many other unexpected applications of his work have been found in classical number theory.

Curiously Lindenstrauss turned 40 on the 1st of August. I am not sure if there have been any other Fields medalists who were over 40 on the day of the award. Presumably the official cut-off date for the 40 year age limit is earlier.

A Fields Medal for Cedric Villani

August 19, 2010

Cedric Villani has worked as a mathemtical physicist on kinetic theory and especially the Boltzmann equation and its variants. It may come as a surpirse that new and important results can be discovered in this area that dates from the nineteenth century, but that is indeed the case.

Villani is a French mathematician who studied at the Ecole Normale Superieure. He is now the director of the Institut Henri Poincare in Paris, His best known work is on optimal transportation theory and convergence to equilibrium.

A Fields Medal for Ngô Bảo Châu

August 18, 2010

It had been widely predicted that Ngô Bảo Châu would be awarded a Fields medal this year. The reason was his proof of  the Fundamental Lemma for the case of reductive groups last year. This is considered a breakthrough step in the Langlands Program, a collection of far-reaching conjectures linking number theory and group representation theory.

Ngô Bảo Châu was born in Hanoi in 1972, making this the last time he would be eligible for the Fields Medal before passing his 40th birthday. Like many field medalists his talent for maths became apparent early. He was a double gold medalist at the International Maths Olympiads with a rare perfect score in one year. After becoming the youngest professor in Vietnam in 2005 he moved to the US and is now at the IAS in Princeton.

in 2004 Ngô Bảo Châu proved the fundamental lemma for unitary groups with Gérard Laumon, but it is his more general result from 2008 when he proved the lemma for all reductive groups that made him a mathematical legend. This is just the kind of singular achievement that the Fields medal likes to recognise.

A result in mathematics is called a lemma if it is a step required towards proving a bigger and more important theorem. Typically a lemma is easier to prove but in the case of the fundamental lemma this was far from true. In the 1960s, Robert Langlands had found a strategy for proving a set of important results known now as the Langlands conjectures, relating Galois theory from algebra to automorphic forms in analysis.  The proves were incomplete and one step needed to bridge the gaps was to show that certain identities on reductive groups hold. This was the lemma that Ngô Bảo Châu finally proved. As a consequence many important relationships of the Langlands program were then also known to hold. Mathematician Peter Samak described the feeling of the those working in the field by saying “It’s as if people were working on the far side of the river waiting for someone to throw this bridge across. And now all of sudden everyone’s work on the other side of the river has been proven.”

The Langlands program remains one of the hardest areas of mathematics to penetrate because it requires expertise in different branches from number theory to algebra, special functions and algebraic geometry. Since Langlands initial insights it has gradually become clear that the conjectures are important to understanding of many beautiful results. They are even thought by some to have deep applications in theoretical physics.