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Home https://server7.kproxy.com/servlet/redirect.srv/sruj/smyrwpoii/p2/ Science https://server7.kproxy.com/servlet/redirect.srv/sruj/smyrwpoii/p2/ Karen Uhlenbeck, Uniter of Geometry and Analysis, won the Avel Award

Karen Uhlenbeck, Uniter of Geometry and Analysis, won the Avel Award



Karen Ulebbeck, whose initial work helped open a whole field of mathematics, was awarded to Abel at a ceremony today in Oslo. The prize, created by the Norwegian Academy of Science and Letters in 2003, is widely perceived as a mathematical equivalent of the Nobel Prize.

Her study "inspires a generation of mathematicians," says François Laboury of the University of the Cote d'Azur in France. "He traverses and discovers new things that no one has ever discovered." Ulenbeck, who was born in Cleveland in 1942, was an insatiable reader as a child, but she was not interested in mathematics until she enrolled in the first math at the University of Michigan. "The structure, elegance and beauty of mathematics struck me immediately and I lost my heart to her," she wrote in the book The Mathematicians: An External View of the Inner World . a feature that attracted her at that time: this is something you can work on loneliness if you want. In her early life, she said in 1

997, "I thought something about people was a terrible profession."

In the mid-1960s, Ulenbeck attended a college campus at Brandeis University, where she chose Richard Palle. advisor. Palais then explored an unexplored territory between analysis (summation of calculus) and topology and geometry (studying the structure of the figures). "I was attracted to this – the field between things," said Ulenbek in an interview last year at Celebratio Mathematica . "It was like jumping off the deck where he did not know what was going to happen."

Pal and mathematician Stephen Smile (who won a field medal for his topological survey shortly thereafter) has just made the headline on harmonious maps, "which will be a springboard for some of Ullenbek's most important results. The study of harmonic maps can be traced to the centuries-old field of mathematics, called alternate calculus, which seeks forms that are in equilibrium with some natural physical measurements such as energy, length, or area. For example, one of the oldest and most well-known problems in calculating variance is the brahisto-crown problem posed by Johann Bernoulli in 1696, which requires the curve that the ball will roll the fastest from one point to another. 19659008] To understand what a card means to be harmonious, imagine a compact shape made of rubber – rubber band, say, or rubber sphere. Then choose a specific way to place this shape in a space (for example, an infinite three-dimensional space or a three-dimensional donut form). This positioning of the shape is called a harmonic map if, roughly speaking, it puts the shape in equilibrium, meaning that the rubber will not break into any different configuration that has a lower elastic potential energy (what mathematicians call energy Dirichle). When the space in which you map the rubber form is a complex object with holes (such as a donut surface or its higher dimensions), different harmonic maps may appear. For example, if you wrap a rubber band around the central opening on the donut surface, the tape can not shrink to a point without coming out of the donut surface – instead it will shrink to the shortest path around the hole.

When the rubber form or the target space is some intricate, perhaps a high-dimensional object, understanding the range of harmonic mapping options can be difficult because we can not just build a physical model and then see what the tire is doing. Intuitively, we can try to build a harmonic map by starting with each map and then looking for ways to deform it, little by little, to bring it to equilibrium. But it is not always clear whether this process will eventually converge and reach equilibrium.

This question makes sense for other energy measures besides Dirichle's energy, while Palais and Smail have provided for energy measures that, when implemented, ensure that at least some of these deformation processes really get closer together. The condition of Palais-Smale was perfect for the one-dimensional case of harmonic maps, where we compare (infinitely narrow) rubber band into a compact space such as a sphere or a donut surface. But when the rubber shape has a dimension greater than one – if it is a surface, for example, or some high-dimensional object – Dirichle's energy does not always satisfy the condition of Palais and Smaill, which means that the process of gradual deformation of mapping to reduce of his Dirichle Energy sometimes fails to get close to a harmonic map. Her five years at Urbana-Champaign were not particularly happy – she and her then husband were professors there, and she felt as if she had been seen mostly as a "teaching wife" – but while she met Jonathan Sachs, post-graduate student, Together they studied you a series of different energy measures on two-dimensional surfaces that satisfy the condition of Palle-Smeal and which approximate Dirichle's energy. For each of these alternative energy measures, the Palais-Smale condition ensures that there is a minimal energy card. As the energy in question approaches nearer to Dirichle's energy, do Ulenbek and Sachs ask whether these cards are approaching a harmonic map?

The answer they showed in the late 1970s and early 1980s was "almost". almost any point on the surface, these maps are collected to a harmonic map. But with a limited collection of dots on the surface, cards can begin to form a very specific type of singularity called bubble where there is no way to make sense of mapping. chew again chewing gum, and you blow a bubble but then gradually pull more and more gum into your mouth while keeping the bubble of the same size. The bubble-forming arc will stretch thinner and thinner, but the bubble will remain viable during this process (at least in an idealized environment where the tire stretches infinitely). But at the end of this process, the bubble will pop up after you've removed the whole tire in your mouth.

Similarly, Sacks and Uhlenbeck have shown that maps that minimize alternative forms of energy approximate to a harmonic map almost anywhere, but close to several points on the surface that begin to form bubbles. With the approaching and approaching of energy to Dirichle's energy, these bubbles will be built up of smaller and smaller spots on the surface. At the end of the process, when we reach Dirichle's actual energy, the card will want to make a whole bubble from a single point, so we'll hit a peculiarity. Since compact space makes the surface a thing – like a sphere or surface of a donut – there are only a limited number of holes around which bubbles can form, there are only a finite many of these pulsating singularities. , B, e, V, N, T, S)
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