Classification Problems

分类问题

• 批准号: 9970403
• 负责人: Greg Hjorth
• 金额: $ 12.01万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-08-01 至 2003-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9970403&HistoricalAwards=false
• 关键词: Classification; Problems

9970403Hjorth Since many mathematical objects arise as the quotients of someconcrete space by an equivalence relation, attempts to classify suchobjects leads naturally to the study of equivalence relations on Polishspaces and the reducibility order between these equivalence relations. Inparticular, Hjorth's project addresses structural issues about the Borelequivalence relations in which every equivalence class is countable, andorbit equivalence relations on Polish spaces arising from the continuousactions of Polish groups somewhat more complicated than the infinitesymmetric group. In the first case, there is a poverty of techniques fordistinguishing the Borel cardinalities of the quotient spaces, and in thesecond, he would hope to extend the methods from model theory that havebeen so wildly successful in the case of the infinite symmetric group. In a very rough sense, one might try to divide mathematics into thepart that seeks to determine some quantity or equation and the part thatseeks simply to understand some abstract class of mathematical objects.In the former part one has many examples, ranging from areas of appliedmathematics through to the recent proof of (the inequality comprising)``Fermat's last theorem.'' In the second part one has certain areas inabstract analysis (for instance, the classification of commutative C-staralgebras), or even certain parts of topology (the classification of two-dimensional manifolds), or algebra (for instance, perhaps, the Heckealgebras, which played such an important part in Wiles's proof), and thevery abstract kinds of objects that can appear there. Here the intentionis that simply understanding these objects can lead to unforseeable advances.Hjorth's project concerns this second kind of mathematics, and it might beloosely described as concerned with which kinds of objects from onemathematical specialty can in principle be reduced to which other kinds ofobjects from other subspecialties.***

9970403Hjorth由于许多数学对象通过等价关系而成为体形结合空间的商，因此试图对此类对象进行分类自然会导致在抛光范围上的等价关系的研究以及这些等价关系之间的降低性顺序。 hjorth的内部项目解决了有关硼期关系的结构性问题，在该关系中，每个等效类都是可数的，在波兰人空间上的Andorbit等效关系是由于波兰群体的连续行动而引起的，而不是Infitientientymmememmetric群体。 在第一种情况下，技术存在贫困，以赋予商人的骨基础性，而在第三种方面，他希望将方法从模型理论中扩展出来，这些方法在无限对称群体的情况下如此成功。 从非常粗略的意义上讲，人们可能会试图将数学划分为试图确定一些数量或方程式的部分，而索取的部分只是为了理解一些抽象的数学对象类别。实例，可交换c stargebras的分类，甚至是拓扑的某些部分（二维流形的分类）或代数（例如，也许，在Wiles的证明中起着如此重要的作用），以及可能出现在Wiles的证明中的重要作用），以及可能出现在那里的任何抽象物体。 这里的意图是简单地理解这些物体可能会导致无法实现的进步。Hjorth的项目涉及第二种数学，并且可能会被描述为关注哪些onemecrathemathemathematical专业的物体原则上可以将其他类型的其他类型的对象降低到其他spepecialties的其他类型。