Two Australian National University mathematicians set out to refine what was thought to be the solution of a problem critical to fields ranging from astrophysics through medicine to finance. They wound up solving the problem almost 200 years after it was formulated.

And the two, Neil Trudinger and Xu-Jia Wang, have recently won prestigious awards for their wider body of work in an area of mathematics at the centre of an international research effort.

Trudinger and Wang were working on the important Monge-Ampere equation, studied by the French mathematician Gaspard Monge in the eighteenth century, and later by another French mathematician - and physicist - Andre-Marie Ampere, a pioneer of the theory of electromagnetism.

The equation belongs to a class of mathematical relationships called second order elliptic partial differential equations. These equations are fundamental to many applications and have recently been the subject of intense study because they describe "transportation problems", involving the flow of matter.

"The Monge-Ampere equation has widespread applications," says Wang. They include the optimisation of public transport systems, tracking the flow of blood through the body, working out the distribution of matter in the universe and tracing the flow of money in financial markets.

It was thought that the associated Monge problem had been solved in the 1970s but it was later discovered that there were gaps in the solution. The Monge problem is a special case of an optimal transportation problem , formulated originally by Monge in 1781 when he posed the problem of moving a heap of dirt from one place to another with the least amount of work. Monge later became Minister for the Navy under Napoleon "Our paper was attempting to simplify it but, because of the flaws, we were, in fact, solving it for the first time," says Trudinger, a professor in theoretical mathematics, and a Fellow of Australian Academy of Science and of the Royal Society of London. An American team also came up with a solution about the same time.

Trudinger has won the American Mathematical Society's Leroy P. Steele Prize for Mathematical Exposition for his research monograph, Elliptic Partial Differential Equations of Second Order, co-authored with colleague, the late David Gilbarg.

Meanwhile, Wang, has won the prestigious 2007 Morningside Gold Medal of Mathematics from the International Congress of Chinese Mathematicians, an honour that ranks him alongside some of the world's leading mathematicians.

Wang immigrated to Australia from China in 1995 to work with Trudinger and colleagues. He has been studying elliptic partial differential equations for 25 years, and admits to thinking about mathematics most of the time.
The theoretical mathematicians often don't hear of new applications of their work until years after they have published it. In this case, due to the breadth of applications of the Monge-Ampere equations Wang has already used some of their ideas for a new algorithm for antenna design.