THE MATHEMATICIAN YOU DON’T KNOW

March 19, 2019


Okay, there will be a test after you read this post.  Here we go.  Do you know these people?

  • Beyoncé
  • Jennifer Lopez
  • Mariah Cary
  • Lady Gaga
  • Ariana Grande
  • Katy Perry
  • Miley Cyrus
  • Karen Uhlenbeck

Don’t feel bad.  I didn’t know either.  This is Karen Uhlenbeck—the mathematician we do not know.  For some unknown reason we all (even me) know the “pop” stars by name; who their significant other or others are, their children, their latest hit single, who they recently “dumped”, where they vacationed, etc. etc.  We know this. I would propose the lady whose picture shown below has contributed more to “human kind” that all the individuals listed above.  Then again, that’s just me.

For the first time, one of the top prizes in mathematics has been given to a woman.  I find this hard to believe because we all know that “girls” can’t do math.  Your mamas told you that and you remembered it.  (I suppose Dr. Uhlenbeck mom was doing her nails and forgot to mention that to her.)

This past Tuesday, the Norwegian Academy of Science and Letters announced it has awarded this year’s Abel Prize — an award modeled on the Nobel Prizes — to Karen Uhlenbeck, an emeritus professor at the University of Texas at Austin. The award cites “the fundamental impact of her work on analysis, geometry and mathematical physics.”   Uhlenbeck won for her foundational work in geometric analysis, which combines the technical power of analysis—a branch of math that extends and generalizes calculus—with the more conceptual areas of geometry and topology. She is the first woman to receive the prize since the award of six (6) million Norwegian kroner (approximately $700,000) was first given in 2003.

One of Dr. Uhlenbeck’s advances in essence described the complex shapes of soap films not in a bubble bath but in abstract, high-dimensional curved spaces. In later work, she helped put a rigorous mathematical underpinning to techniques widely used by physicists in quantum field theory to describe fundamental interactions between particles and forces. (How many think Beyoncé could do that?)

In the process, she helped pioneer a field known as geometric analysis, and she developed techniques now commonly used by many mathematicians. As a matter of fact, she invented the field.

“She did things nobody thought about doing,” said Sun-Yung Alice Chang, a mathematician at Princeton University who served on the five-member prize committee, “and after she did, she laid the foundations for that branch of mathematics.”

An example of objects studied in geometric analysis is a minimal surface. Analogous to a geodesic, a curve that minimizes path length, a minimal surface minimizes area; think of a soap film, a minimal surface that minimizes energy. Analysis focuses on the differential equations governing variations of surface area, whereas geometry and topology focus on the minimal surface representing a solution to the equations. Geometric analysis weaves together both approaches, resulting in new insights.

The field did not exist when Uhlenbeck began graduate school in the mid-1960s, but tantalizing results linking analysis and topology had begun to emerge. In the early 1980s, Uhlenbeck and her collaborators did ground-breaking work in minimal surfaces. They showed how to deal with singular points, that is, points where the minimal surface is no longer smooth or where the solution to the equations is not defined. They proved that there are only finitely many singular points and showed how to study them by expanding them into “bubbles.” As a technique, bubbling made a deep impact and is now a standard tool.

Born in 1942 to an engineer and an artist, Uhlenbeck is a mountain-loving hiker who learned to surf at the age of forty (40). As a child she was a voracious reader and “was interested in everything,” she said in an interview last year with Celebratio.org. “I was always tense, wanting to know what was going on and asking questions.”

She initially majored in physics as an undergraduate at the University of Michigan. But her impatience with lab work and a growing love for math led her to switch majors. She nevertheless retained a lifelong passion for physics, and centered much of her research on problems from that field.  In physics, a gauge theory is a kind of field theory, formulated in the language of the geometry of fiber bundles; the simplest example is electromagnetism. One of the most important gauge theories from the 20th century is Yang-Mills theory, which underlies the standard model of elementary particle physics. Uhlenbeck and other mathematicians began to realize that the Yang-Mills equations have deep connections to problems in geometry and topology. By the early 1980s, she laid the analytic foundations for mathematical investigation of the Yang-Mills equations.

Dr. Uhlenbeck, who lives in Princeton, N.J., learned that she won the prize on Sunday morning.

“When I came out of church, I noticed that I had a text message from Alice Chang that said, Would I please accept a call from Norway?” Dr. Uhlenbeck said. “When I got home, I called Norway back and they told me.”

Who said women can’t do math?

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