Some Problems on the Edge of Descriptive Set Theory

这个项目的细节涉及三个方向，这三个方向是出于对可定义的等价关系的普遍兴趣。这些方向中的第一个涉及树形等价关系。根据Adams和Kechris的工作，我们知道存在大量相互不可比较的可数Borel等价关系。对于可树的Borel等价关系，还没有这样的结果。我们不知道是否有无限多个不同的例子，我们基本上只有一个确定的例子，它不是超有限的。更广泛地说，我们不知道涉及自由群的自由行为的测度论例子的隐含牵涉是否是超有限的唯一障碍。Hjorth项目的第二个方向涉及到量词自由逻辑拓扑学中可数同构类型的Borel复杂性的精细研究问题，并可能与一些基本概念从一阶逻辑到量词自由上下文的翻译有关。建议的第三个方向是研究一些组合问题，例如对于可定义的划分，对于无限语句，特别是那些作为某些可数结构的Scott语句出现的组合问题，具有一定的划分性质；这可能与谢拉在1970年的《S》一书中提出的关于可数无限逻辑的汉夫数到连续统的一个仍然悬而未决的问题有关。这个领域大约在19世纪末出现，是为了更好地理解微积分中出现的基本对象--如实数线、实值函数、实数的子集、二维和三维空间的子集或区域、这些子集的面积或体积--的努力的一部分，这些对象在工程、物理和微分方程中的应用是必要的。描述性集合论本身并没有解决这些最终应用中的任何一个，而是相当专注于纯粹的基础问题。继1970年S的Silver定理之后，许多描述性集合论者对空间上的等价关系产生了兴趣，如实数线、二维空间或类似的“拓扑空间”类。对这种等价关系的研究导致了商对象的产生，这些商对象是通过考虑所有等价类的集合而产生的。例如，如果我们将两个实数设为等价，而从另一个减去另一个的结果是一个整数(即“整数”)，那么等价类的集合可以用基本上将实数线环绕其自身的结果来有利地识别，以获得一个圆周。在这个简单的例子中，商对象很容易理解，并且具有几何表示。Hjorth领域的大部分工作涉及所谓的“非光滑”等价关系，其商对象不允许这样的表示，而对这些商空间的研究被认为与各种数学学科有关，如“动力学”、“遍历理论”，以及“分析”的一些更抽象的分支，如“无限维群表示”。

The specifics of this project concern three directions whicharose out of a general interest in definable equivalencerelations. The first of these directions relates to the treeableequivalence relations. Following work of Adams and Kechris, weknow that there is a mass of countable Borel equivalencerelations which are mutually incomparable. No such result isknown for the treeable Borel equivalence relations. We do notknow whether there are infinitely many distinct examples, and webasically have only one established example which is nothyperfinite. More generally we do not know whether the implicitinvolvement of measure theoretic examples involving free actionsof the free group is the sole obstruction to hyperfiniteness. Thesecond direction of Hjorth's project concerns issues in the finestudy of Borel complexities of countable isomorphism types in thetopology of quantifier free logic, and may be connected with atranslation of some basic concepts from first order logic into aquantifier free context. The third direction of the proposal isto investigate some combinatorial questions, such as having amodel with a certain partition property for definable partitions,for infinitary sentences, especially those arising as the Scottsentence of some countable structure; this may be related to astill open problem posed by Shelah in the 1970's on the Hanfnumber up to the continuum for countably infinitary logic.In very general terms, this project can be located inside thebranch of mathematics known as "descriptive set theory". Thisarea arose around the end of 19th century as part of an effort tobetter understand the basic objects -- such as the real numberline, real valued functions, subsets of the reals, subsets orregions of two dimensional and three dimensional space, the areaor volume of such subsets -- which appear in calculus, and whichare needed for applications in engineering, physics, anddifferential equations. Descriptive set theory does not itselfactually address any of these eventual applications, but israther preoccupied with purely foundational issues. FollowingSilver's theorem in the 1970's, many descriptive set theoristshave become interested in equivalence relations on spaces such asthe real number line, or two dimensional space, or similarclasses of "topological spaces". The study of such equivalencerelations leads to quotient objects which arise by consideringthe collection of all equivalence classes. For instance if weset two real numbers to be equivalent when the result ofsubtracting one from the other is an integer (i.e. a "wholenumber"), then the collection of equivalence classes may benaturally identified with the result of basically wrapping thereal number line around itself, to obtain circle of circumferenceone. In this simple example the quotient object is easilyunderstood, and has a geometrical representation. Most of thework in Hjorth's area deals with the so called "non-smooth"equivalence relations whose quotient objects do not admit such arepresentation, and the study of these quotient spaces is knownto have connections with a variety of mathematical disciplines,such as "dynamics", and "ergodic theory", and some of the moreabstract branches of "analysis", such as "infinite dimensionalgroup representations".