Classification Problems

由于许多数学对象是作为某一具体空间的商而产生的等价关系，对这些对象进行分类的尝试自然导致了对Polish空间上的等价关系以及这些等价关系之间的可约化阶的研究。特别是，Hjorth的项目解决了Borele等价关系的结构问题，其中每个等价类都是可数的，并且在波兰空间上的轨道等价关系产生于波兰群的连续行动，这些群比无限对称群稍微复杂一些。在第一种情况下，缺乏区分商空间的Borel基数的技术，而在第二种情况下，他希望推广模型理论中的方法，这些方法在无限对称群的情况下非常成功。在非常粗略的意义上，一个人可能试图将数学分为寻求确定某个量或方程式的部分和试图简单地理解某种抽象的数学对象的部分。在前一部分中，有许多例子，从应用数学的领域到最近对“费马最后定理”的证明(包含的不等式)。在第二部分中，人们有抽象分析的某些领域(例如，交换C-星代数的分类)，甚至是拓扑学的某些部分(二维流形的分类)，或者代数(例如，可能在Wiles的证明中起到如此重要作用的Hecke代数)，以及可以出现在那里的非常抽象的对象种类。这里的意图是，仅仅理解这些对象就可以带来不可预见的进步。Hjorth的项目涉及第二种数学，它可能被恰当地描述为关注原则上来自一个数学专业的哪些类型的对象可以简化为来自其他子专业的其他类型的对象。

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.***