Some Problems on the Edge of Descriptive Set Theory

描述集合论边缘的一些问题

• 批准号: 0140503
• 负责人: Greg Hjorth
• 金额: --
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-06-01 至 2008-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0140503&HistoricalAwards=false
• 关键词: Some; Problems; Edge; Descriptive; Set

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".

该项目的具体细节涉及三个方向，这些方向是出于对可定义的等价关系的一般兴趣而产生的。这些方向中的第一个涉及三级等效关系。 在亚当斯（Adams）和凯奇里斯（Kechris）的工作之后，我们知道，有大量可计数的bor骨等效性是无与伦比的。 对于可食用的bor骨等效关系而言，尚无这种结果。我们不知道是否有许多不同的例子，并且在网络上只有一个既定的示例，而不是备受极限的示例。更普遍地说，我们不知道涉及自由群体的自由行动的理论示例的含义是否是唯一的过度障碍。 Hjorth项目的第一个方向涉及在无量词的遗传学中可数字同构类型的Borel复杂性中的问题，并且可能与从一阶逻辑到含水剂免费上下文中的一些基本概念的渗透有关。该提案的第三个方向是研究一些组合问题，例如将Amodel带有特定分区属性，用于定义分区，用于无限句子，尤其是那些作为某些可计数结构的Scottsentence而产生的；这可能与Shelah在1970年代在Hanfnumber上构成的Astill Open问题有关，直到连续体的无限逻辑。 Tobetter努力的一部分是在19世纪末左右出现的。子集 - 在微积分中出现，以及工程，物理和差异方程中应用所需的子集。描述性集理论本身并不是要解决这些最终应用中的任何一个，而是纯粹是纯粹的基础问题。 lasteringsilver的定理在1970年代，许多描述性的定理夏夫对诸如真实数字线或二维空间或“拓扑空间”类似的空间上的等效关系产生了兴趣。对这种等价关系的研究会导致商对象，这些对象通过考虑所有等效类别的收集而产生。 例如，如果Weset两个实数是等效的，而当一个从另一个的结果是一个整数是一个整数（即“ tholenumber”），那么等价类的集合可能会季节性地识别出来，并且基本上包裹自身周围的Thereal数字行的结果，获得周长圆。在这个简单的示例中，商对象很容易被忽略，并且具有几何表示。 Hjorth领域的大部分工作都涉及所谓的“非平滑”等效关系，该关系的商对象不承认这种代表，并且对这些商空间的研究众所周知，可以与各种数学学科（例如“动态”）建立联系。 ，“千古理论”，以及“分析”的某些杂项分支，例如“无限尺寸格式表示”。