Die einflussreiche israelische Tageszeitung Haaretz hatte vorletzte Woche einen langen Artikel über G.D.Mostow, den Entdecker des Mostowschen Starrheitssatzes. Der besagt, dass Mannigfaltigkeiten (mit Ausnahme von Flächen) höchstens bis auf Reskalierung eine lokal-symmetrische Metrik von endlichem Volumen haben, zum Beispiel hyperbolische Metriken endlichen Volumens auf 3-Mannigfaltigkeiten also eindeutig sind. (Man kann dann diese eindeutigen Metriken benutzen, um die Topologie der Mannigfaltigkeiten zu verstehen. Geometrische Invarianten wie z.B. das Volumen sind automatisch auch topologische Invarianten.)

Die Geschichte der Entdeckung des Starrheitssatzes, wie sie in Haaretz beschrieben wird:

“Earlier, in my office, I had been thinking intensively about the problem. I did not think about it while I was driving, but at the light, at that moment, I suddenly thought: ‘Use ergodicity!'”

Ergodizität ist eine Eigenschaft von Bewegungen (oder allgemeiner von Gruppenwirkungen), sie besagt, dass alle Bewegungsbahnen (und auch die Vereinigungen beliebig vieler Bewegungsbahnen) entweder (bis auf eine Nullmenge) den ganzen Raum ausfüllen oder aber selbst Nullmengen sind:

Die Geschichte mit der roten Ampel erinnert ein wenig an Poincarés Entdeckung der automorphen Funktionen.

Der Artikel (Anlaß war die Verleihung des Wolf-Preises Anfang Mai) schreibt sehr interessant über Mostows Biographie, bleibt aber sehr im Allgemeinen, wenn es um Mathematik geht. (Zitat: For the benefit of the general public the committee wrote that “In Mostow’s work one finds a stunning display of a variety of mathematical disciplines,” and that “few mathematicians can compete with the breadth, depth and originality of his works.”)

Um hier noch kurz etwas zu der im Haaretz-Artikel fehlenden Mathematik zu sagen:

Mostows Starrheitssatz besagt im einfachsten Fall folgendes: wenn M und N geschlossene hyperbolische Mannigfaltigkeiten der Dimension >2 sind und eine statige Abbildung f:M–>N einen Isomorphismus der Fundamentalgruppen induziert, dann gibt es eine Isometrie M–>N. (Die Isometrie ist dann sogar homotop zu f. Ein entsprechender Satz gilt allgemeiner für Mannigfaltigkeiten endlichen Volumens, das wurde aber nicht von Mostow, sondern von Prasad bewiesen.) Insbesondere gibt es (bis auf Isometrie) nur eine hyperbolische Metrik auf einer Mannigfaltigkeit der Dimension >2. ( In Dimension 2 gilt der Starrheitssatz nicht, dort gibt es den 6g-6-dimensionalen Teichmüller-Raum, der die hyperbolischen Metriken auf einer Fläche vom Geschlecht g klassifiziert.)

Die Beweisidee ist folgende: die Abbildung f ist auf jeden Fall eine Quasi-Isometrie (wegen Kompaktheit von M), man bekommt also eine bzgl. der Wirkungen der Fundamentalgruppen äquivariante Quasi-Isometrie zwischen den universellen Überlagerungen, also vom hyperbolischen Raum auf sich selbst. Auf dem Rand des hyperbolischen Raumes induziert das eine äquivariante quasi-konforme Abbildung. Wenn man beweisen könnte, dass sie konform ist, dann wäre sie die Randabbildung einer äquivarianten Isometrie und man könnte dann leicht beweisen, dass die ursprüngliche Abbildung zu dieser Isometrie äquivariant homotop ist.

Die Idee, welche Mostow vor der roten Ampel gekommen sein soll, ist nun, Ergodizität zu benutzen. Die Wirkung der Fundamentalgruppen auf dem Rand im Unendlichen ist ergodisch (das folgt mit einem bekannten Beweis daraus, dass die Quotientenmannigfaltigkeit kompakt ist.) Man kann zeigen, dass sogar die Wirkung auf der Menge disjunkter Punktepaare des Randes im Unendlichen ergodisch ist. Daraus wiederum folgt, dass jede auf Punktepaaren definierte stetige Funktion konstant sein muss. Das kann man dann wie folgt anwenden. Die quasikonforme Funktion ist fast überall differenzierbar und falls sie nicht konform ist, kann man in jedem Punkt die Richtung betrachten, in der die Ableitung maximal wird. Zu je zwei Punkten kann man diesen Richtungsvektor entlang der eindeutigen Verbindungsgeodäte aus dem einen Punkt in den anderen verschieben und dann den Winkel mit der dortigen Richtung maximaler Ableitung bilden. Das definiert eine Funktion auf Punktepaaren, die wegen der Ergodizität konstant sein muss. Letzteres führt aber zu einem Widerspruch, weshalb die Ableitung und dann auch die Funktion selbst konform gewesen sein muss.

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Kommentare (5)

  1. #1 Thilo
    27. Mai 2013

    Der Artikel scheint inzwischen hinter einer Abo-Wall zu sein. Ich hoffe, man sieht es mir nach, wenn ich ihn hier ausnahmsweise kopiere:

    Enlightenment at a red traffic light
    Wolf Prize laureate Prof. George Daniel Mostow made his greatest scientific breakthrough while driving.
    By Ofer Aderet | May.12, 2013 | 6:05 PM

    The attempt to explain why Prof. George Daniel Mostow was awarded the Wolf Prize in mathematics last week was doomed to failure. If you are not a certified mathematician, you probably will not understand the significance of his most famous achievement: the discovery of the rigidity phenomenon in geometry, known as the strong rigidity theorem.

    The explanation supplied by the Wolf Prize committee does not make it easier for the average person to understand the greatness of his work. The committee members wrote that Mostow “established a deep connection between continuous and discrete groups, or equivalently, a remarkable connection between topology and geometry.”

    The anonymous judges added another, no less complicated reason for awarding Mostow the prestigious prize: his “work on examples of nonarithmetic lattices in two and three dimensional complex hyperbolic spaces.” For the benefit of the general public the committee wrote that “In Mostow’s work one finds a stunning display of a variety of mathematical disciplines,” and that “few mathematicians can compete with the breadth, depth and originality of his works.”

    Last week, two months before his 90th birthday, the Jewish mathematician arrived in the Holy Land to receive the prize from Israel’s President Shimon Peres. Mostow took the opportunity to meet with family and to celebrate another event: “My birthday is on July 4, but I’ve received permission to declare I am 90 years old two months before the official date,” he said at the end of the interview, minutes before setting out for a swim off the Tel Aviv shore.

    The Wolf Prize is awarded annually in Israel to scientists and artists from abroad, for “achievements in the interest of mankind and friendly relations among peoples.” So far it has been awarded to 272 laureates from 23 countries, for their contributions to agriculture, chemistry, mathematics, medicine, physics, architecture, music, painting and sculpture.

    Mostow is the oldest laureate to come to Israel to receive the prize. “Few can compete with the depth and originality of his works,” explains Mina Teicher, Professor of Mathematics at Bar-Ilan University and a member of the Wolf Foundation council. “There’s something a bit amazing mathematically in his work, but also in his life in general.”

    Mostow’s biography is unusual not only because of his scientific achievements. He made his most significant discovery at the age of 50. In the world of mathematics, where many scientists gain renown for important discoveries at a very early age, this is atypical.

    “They say that mathematics belongs to the young and that the major contributions in the field are made before the age of 30,” says Mostow. “In my case that’s not quite so. I worked on a very difficult problem, which engaged the minds of excellent researchers, until the point at which they reached a dead end. It took me until the age of 50, but finally I saw a way of overcoming the problem.”

    In an interview with the Yale University website, Mostow said that the solution to the problem came to him while waiting at a red light in New Haven, Connecticut, where he has resided since the 1960s. “Earlier, in my office, I had been thinking intensively about the problem. I did not think about it while I was driving, but at the light, at that moment, I suddenly thought: ‘Use ergodicity!’ [Ergodicity is a concept in probability and statistics.] When I got home, I immediately sketched out a path to the theorem. In fact, I only had to add eight pages to what I already had done to complete the proof.”

    Since then, he says, “I get a high every time I pass that intersection.”

    Married at 84

    His scientific discovery is not the only thing Mostow achieved at a late age. He also found his second love at an unusual age, after the death of his first wife. One day before his 84th birthday he married Sidnie, a mathematician herself and the widow of his friend and colleague at the Department of Mathematics in Yale University. His wife announced their intention to marry to his family by e-mail.

    He was born in Boston to Jewish parents who immigrated to the United States from Ukraine in the early 20th century. As a child he studied Hebrew, which he speaks to this day. “If I was to stay in Israel now for a month or two, I could speak Hebrew fluently,” he says in slightly flawed Hebrew. “As a boy I studied all subjects in school in Hebrew – literature, Talmud, Bible, Torah. We read the History of the Jewish People [By Simon Dubnow]. I have quite a large vocabulary, but I cannot conduct a simple conversation.”

    From here on Mostow shifts back to English. “My world was filled with Jews. I knew no one else,” he recalls.

    One of his earliest memories is from the age of four. “I was a shy boy. The teacher decided that all the children beside me would graduate to the next class. My mother went to the principal, who gave me a test. I had to say which objects were missing in the picture.”

    He passed the test easily, but his love for mathematics only blossomed in high school. “I especially enjoyed challenging problems. But I did not know that mathematics was a profession. I am indebted to my high school English teacher who, in my senior year, called me up to his desk to ask about my career plans, and told me that his brother was a mathematician. I decided then and there that mathematics was for me.”

    World War II broke out when he was in high school. His brother Jechiel, only a year and two months older than him, was drafted into the U.S. army and served as a tail gunner in the U.S. Air Force. At the end of 1942, when he was 18, his plane was shot down by German fire over the Gulf of Tunis. At first Mostow’s mother hoped that her son, defined as “missing in action,” would eventually return. “It was a period of major depression. My mom took years to recover,” he recalls.

    When it transpired that there was no chance Jechiel would return alive, Mostow named one of his sons after him.

    In 1948 Mostow completed his Ph.D. dissertation for Harvard University. Later he worked at Johns Hopkins University and at Yale, where he remained until retiring in 1999. In the 1970s he was elected to the National Academy of Science. In the late 1980s he served as the president of the American Mathematical Society.

    Not scared of Math

    During the Cold War he was one of the few American scientists to maintain close relations with Soviet colleagues, at a time when the political atmosphere was very tense. He used his connections to help Russian mathematicians out to the West and save them from the Communist regime. “We felt a sense of injustice, since talent was not the central requisite for scientific advancement in the Soviet Union. In addition, there were different tests for Russian and Jewish scientists,” he recalls.

    Mostow decided to make use of his senior status at the Mathematical Society and speak out. “I had to make a decision: ignore what we were seeing just to ‘keep the peace,’ hoping things would get better in the future, or speak out about it,” he says.

    In the early 1980s an international congress of mathematicians was supposed to convene in Warsaw. “Mathematicians were imprisoned there for political reasons. We decided to announce that the congress would be postponed until their release. Within a short time they were released. In 1983 the congress took place, with great success, both scientifically and politically.”

    The tough Communist regime also had other aspects, such as an unexpectedly beneficial influence on mathematics. “People avoided professions such as political science, which was considered dangerous. They preferred studying math, which was safe. In general, the Russians developed a math education system which we can envy to this day,” he says.

    For this reason, he says, to this day in Russia, unlike in other countries, “people aren’t afraid of math.”

    A phobia of math, as he calls it, is rampant in the West, where “teachers don’t know the profession and don’t understand what they’re teaching, and they transfer that fear to students to the level of an epidemic. We need good math teachers, to train skilled mathematicians in all fields. True, just as not everyone who plays the piano becomes a pianist, not everyone who studies math becomes a mathematician. But in order to enable proper development, we need good teachers.”

    Mostow has four children, 10 grandchildren and 14 great-grandchildren. When asked about his connection with Israel, even the genius mathematician has to do some math. “Just a minute – I need to count how many of them live in Israel,” he says. The following sentence sound like a math riddle: “All my great-grandchildren are the grandchildren of my son Mark, who is the only one of my four children to live in Israel. So do all three of his children. So I have three grandchildren and 14 great-grandchildren in Israel.”

    After a short pause he adds that this summer the equation will change. The daughter of another son of his “plans to make aliya after she gets married.”

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