Ratner's theorems and applications

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Wen-Liang Tseng , National Taiwan University

In the early 1980s, Marina Ratner published three papers about horocycleÌýflows on the unit tangent bundle of a surface of constant negative curvatureÌýwith finite volume. In the early 1990s, by expanding the ideas from herÌýstudy of horocycle flows, Ratner proved a series of beautiful results aboutÌýunipotent flows on homogeneous spaces. These are Ratner MeasureÌýClassification Theorem, Ratner Orbit Closure Theorem, and RatnerEquidistribution Theorem. These theorems are not only beautiful inÌýtheir own but have far-reaching influence until today. For example, theÌýOppenheimÌýConjecture proven by Gregory Margulis in 1987 can be viewed as a specialÌýcase of Ratner Orbit Closure Theorem. The work about Arithmetic QuantumÌýUnique Ergodicity proven by Elon Lindenstrauss in 2006 applied RatnerÌýMeasure Classification Theorem and the Shearing property-a crucial part ofÌýRatner's proof. In this talk, these theorems, the Shearing property, and howÌýthey are used in the proof of Oppenheim Conjecture will be introduced. IfÌýtime permits, we will show how to use the Shearing property to getÌýfiniteness of fibers of a quotient map.