Classification Problems

9970403Hjorth由于许多数学对象是通过一个等价关系作为某个具体空间的商而产生的，所以试图对这些对象进行分类，自然会导致对波兰空间上的等价关系和这些等价关系之间的可约顺序的研究。特别是，Hjorth的项目解决了关于borel等价关系的结构问题，其中每个等价类都是可数的，以及波兰空间上由波兰群的连续作用产生的轨道等价关系，这比无限不对称群更复杂。在第一种情况下，区分商空间的Borel基数的技术贫乏，在第二种情况下，他希望从模型理论中扩展方法，这些方法在无限对称群的情况下已经取得了如此巨大的成功。粗略地说，我们可以试着把数学分为两部分，一部分是试图确定某个数量或方程，另一部分是试图简单地理解一些抽象的数学对象。在前一部分中，有许多例子，从应用数学领域到最近的证明（包含费马最后定理的不等式）。“在第二部分中，我们有抽象分析的某些领域（例如，交换c星代数的分类），甚至拓扑的某些部分（二维流形的分类），或代数（例如，也许，Heckealgebras，它在Wiles的证明中发挥了如此重要的作用），以及可以出现在那里的非常抽象的对象。这里的意图是，简单地理解这些对象可以导致不可预见的进步。霍约斯的研究项目涉及第二种数学，粗略地说，它关注的是来自某一数学专业的哪些对象原则上可以归结为来自其他子专业的哪些对象

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.***