Ratner's theorems and applications

Ratner's theorems and applications

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Wen-Liang Tseng , National Taiwan University
Fine Hall 601

In the early 1980s, Marina Ratner published three papers about horocycle flows on the unit tangent bundle of a surface of constant negative curvature with finite volume. In the early 1990s, by expanding the ideas from her study of horocycle flows, Ratner proved a series of beautiful results about unipotent flows on homogeneous spaces. These are Ratner Measure Classification Theorem, Ratner Orbit Closure Theorem, and RatnerEquidistribution Theorem. These theorems are not only beautiful in their own but have far-reaching influence until today. For example, the Oppenheim Conjecture proven by Gregory Margulis in 1987 can be viewed as a special case of Ratner Orbit Closure Theorem. The work about Arithmetic Quantum Unique Ergodicity proven by Elon Lindenstrauss in 2006 applied Ratner Measure Classification Theorem and the Shearing property-a crucial part of Ratner's proof. In this talk, these theorems, the Shearing property, and how they are used in the proof of Oppenheim Conjecture will be introduced. If time permits, we will show how to use the Shearing property to get finiteness of fibers of a quotient map.