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 由于许多数学对象是通过等价关系作为某个具体空间的商而出现的，因此对此类对象进行分类的尝试自然会导致对波兰空间上的等价关系以及这些等价关系之间的可约化顺序的研究。 特别是，Hjorth 的项目解决了有关 Borelequivalence 关系的结构问题，其中每个等价类都是可数的，以及波兰空间上的轨道等价关系，这些关系是由波兰群的连续作用引起的，这些波兰群比无限对称群更复杂。 在第一种情况下，区分商空间的 Borel 基数的技术很匮乏，而在第二种情况下，他希望扩展模型论中的方法，这些方法在无限对称群的情况下非常成功。 从非常粗略的意义上讲，人们可能会尝试将数学分为寻求确定某些数量或方程的部分和寻求简单地理解数学对象的某些抽象类别的部分。在前一部分中，有很多例子，范围从应用数学领域直到最近证明（包含不等式）“费马大定理”。在第二部分中，人们在抽象分析中具有某些领域（例如，交换 C 星代数的分类），甚至拓扑学（二维流形的分类）或代数（例如，赫克代数，它在怀尔斯的证明中发挥了如此重要的作用）的某些部分，以及可以出现在那里的非常抽象的对象类型。 这里的目的是简单地理解这些对象可以带来不可预见的进步。Hjorth 的项目涉及第二种数学，它可能被宽泛地描述为涉及来自一个数学专业的哪些类型的对象原则上可以简化为来自其他数学专业的其他类型的对象亚专业.***