Karen Uhlenbeck Just Won One of Math's Most Prestigious Prizes. Here's Why Her Work Is So Important.

Live Science | 3/22/2019 | Staff
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U.S. mathematician Karen Uhlenbeck won this year’s Abel Prize, becoming the first woman to take home the prestigious math award, the Norwegian Academy of Sciences and Letters announced March 19.

"I can't think of anyone who deserves it more," said Penny Smith, a mathematician at Lehigh University in Pennsylvania, who has worked with Uhlenbeck and says she has become her best friend. "She really is not just brilliant but creatively brilliant, amazingly creatively brilliant."

Surfaces - Doughnut - Pretzel - Surfaces - Manifolds

Curved surfaces (imagine a doughnut or a pretzel), or even difficult-to-visualize, higher-dimensional surfaces, are generally called "manifolds," Smith said. The universe itself is a four-dimensional manifold defined by a set of partial differential equations, she added.

Uhlenbeck, along with a couple of other mathematicians in the 1970s, developed a set of tools and methods for solving partial differential equations that describe many manifold surfaces.

Work - Uhlenbeck - Jonathan - Sacks - Surfaces

In her early work, Uhlenbeck, along with mathematician Jonathan Sacks, focused on understanding "minimal surfaces." An everyday example of a minimal surface is the outer surface of a soap bubble, which normally settles on a spherical shape because that uses the least amount of energy in terms of surface tension.

But then, say you drop a cube made of wire into a soap solution and pull it back out. The soap still seeks the lowest-energy shape, but this time, it must do so while also somehow clinging to the wire — so, it will form a bunch of different planes meeting at 120-degree angles.

Shape - Soap - Dimensions - Surface - Manifold

Defining the shape of this soap bubble becomes more and more complicated the more dimensions you add, such as a two-dimensional surface sitting in a six-dimensional manifold. Uhlenbeck figured out the shapes that soap films can take in higher-dimensional curved spaces.

Uhlenbeck also revolutionized another area of mathematical physics known as gauge theory.

Here's how it goes. Sometimes when...
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