Computable Structure Theory

奈特和她的学生将致力于可计算结构理论的各个方面。 奈特计划继续与阿基诺一起研究算术的薄弱环节. 她还希望继续与Goncharov和Harizanov，与Young和Shore一起工作，研究一系列相互关联的问题，包括可计算结构的复杂关系，高阶结构和通过Kleene O. 她目前正在与西玛，赫希费尔特，和索尔的总理模型和低性能。 她计划与伦普、麦考伊和所罗门一起研究布尔代数。 这些资金主要用于学生的旅行和书籍。 其余部分是部分支持圣母院逻辑研讨会。 目前，奈特有两个学生。安德鲁·阿拉纳（Andrew Arana）可能在今年夏天毕业，他使用一些新的独立句子（哥德尔-罗瑟句子的变体）在算术复杂性方面有一些结果。 Arana现在专注于更基础的问题。 WesleyCalvert是一名二年级的学生，他正在研究各种熟悉的可计算结构类的同构问题的复杂性。 他已经有了几类领域的成果。 奈特的研究目标，以及与她一起工作的学生的目标，是确定数学结构的哪些方面是可计算的，以及对于那些不可计算的方面，是否存在一些很好的、可计算的近似。 这笔赠款将促进这类新的工作。 有一个更大的目标-使学生成为一流的研究科学家。 这笔赠款将主要用于学生的旅费和书籍。 如果没有钱买书，奈特的学生几乎没有书。 她发现，如果给她的学生钱去买最重要的参考资料，他们读的书会比不这样做时多得多。 他们看到他们正在研究的问题是如何产生的，以及他们的工作对整个领域意味着什么。 奈特的学生和其他逻辑学专业的学生从参加会议的机会中受益匪浅。他们会见了来自其他大学的研究人员，听到了新的发展。 一旦他们有了自己的结果，他们就会进行演讲，并获得宝贵的建议。 简而言之，他们有机会，作为学生，加入社会的研究数学家。

Knight and her students will work on various aspects ofcomputable structure theory. Knight plans to continue work withD'Aquino on weak fragments of arithmetic. She expects also tocontinue work with Goncharov and Harizanov, with Young, and withShore, on a collection of inter-related problems involvingcomplicated relations on computable structures, structures ofhigh rank, and paths through Kleene's O. She is currentlyworking with Csima, Hirschfeldt, and Soare on prime models andlowness properties. She plans to work with Lempp, McCoy, andSolomon on Boolean algebras. The funds are mainly for studenttravel and books. The remainder is for partial support of theNotre Dame Logic Seminar. Currently, Knight has two students.Andrew Arana, who may finish this summer, has a number of resultson complexity in arithmetic, using some new independent sentences(variants of the Godel-Rosser sentence). Arana is nowconcentrating on problems of a more foundational nature. WesleyCalvert, a second year student, is working on complexity of theisomorphism problem for various familiar classes of computablestructures. He already has results for several classes offields. Knight may acquire new students during the period of thegrant.The goal of Knight's research, and that of students working withher, is to determine which aspects of mathematical structures arecomputable, and for those which are not computable, whether thereis some nice, computable approximation. The grant willfacilitate new work of this kind. There is a largergoal---enabling students to become first-rate researchscientists. The grant will be used mainly for student travel andbooks. Without money for books, Knight's students would hardlyown any. She has found that, given money to buy the mostimportant references, her students read much more than they wouldotherwise. They see how the problems they are working on arose,and what their work means for the field as a whole. Knight'spast students, and other logic students at Notre Dame, havebenefited tremendously from opportunities to travel to meetings.They meet researchers from other universities and hear about newdevelopments. As soon as they have results of their own, theygive talks, and they get valuable suggestions. In short, theyhave the opportunity, as students, to join the community ofresearch mathematicians.