Fernando Codá Marques (Princeton University) and André Neves (Imperial College London) will receive the 2016 AMS Oswald Veblen Prize in Geometry for "their remarkable work on variational problems in differential geometry [including] the proof of the Willmore conjecture." The prize citation points in particular to a paper by Codá Marques and Neves, "Min-max theory and the Willmore conjecture," Annals of Mathematics (2014).
This work resolved a longstanding question about the nature of surfaces. The Willmore energy, sometimes called the bending energy, is a formula that describes how surfaces bend in different directions. For example, for a sphere, the Willmore energy is zero, because a sphere bends in exactly the same way at every point. The concept is named after mathematician Thomas J. Willmore (1919-2005), although it was known already in the early 19th century, when it was used for theoretical modeling of elastic shells.
Willmore knew that the sphere minimizes the Willmore energy. In 1965, he began thinking about how the Willmore energy behaves on other surfaces, in particular, on a torus, which is the mathematical name for the surface of a donut with one hole. He conjectured that the Willmore energy of a torus would always be greater than or equal to a certain minimum value. He also found that a particular surface, called the Clifford torus, realizes the minimal Willmore energy.
The Willmore energy is not only of theoretical interest but also arises in nature. In 1991, molecular biologists observed, under a microscope, that fluid-filled sacs in certain cells were Clifford tori. The sacs assumed that shape naturally, to minimize the Willmore energy.
In the decades after 1965, the Willmore conjecture stimulated a great deal of research, because of its naturalness and beauty and because of the richness of the concept of the Willmore energy. But it was not until the 2012 work of Codá-Marques and Neves that the Willmore conjecture was fully resolved. A major breakthrough came when they discovered the essential role played by the key feature of the surfaces in question: they have exactly one hole. The topology of the surfaces ended up playing a strong role in the geometric question the two mathematicians were exploring. In an interview to appear in the Notices of the AMS in the February 2016 issue, Codá Marques said, "It was pure topology, and it was beautiful."
As with many ground-breaking results in mathematics, the work of Codá Marques and Neves has illuminated new approaches to other significant questions, which they are actively pursuing.
Presented annually, the AMS Veblen Prize is one of the highest distinctions for research in topology and geometry. The prize will be awarded on Thursday, January 7, 2016, at the Joint Mathematics Meetings in Seattle.
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