US-SPAIN Cooperative Research: Metric Properties of Thompson's Group

美国-西班牙合作研究：汤普森群的公制性质

• 批准号: 0305545
• 负责人: Sean Cleary
• 金额: $ 1.41万
• 依托单位: CUNY City College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-07-15 至 2007-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0305545&HistoricalAwards=false
• 关键词: US; SPAIN; Cooperative; Research; Metric

The goal of this project is the understanding of the metric properties of Thompson's groups. Thompson defined a fascinating family of groups, which appear in a wide range of mathematical disciplines, including logic, homotopy theory and the measure theory of discrete groups. Thompson's groups are often the simplest or only known examples of a number of strange group-theoretic phenomena. Despite a great deal of study by a number of researchers, many properties of these groups are poorly understood. This project investigates the metric structure of Thompson's groups by developing a better understanding of the Cayley graphs of these groups. This research builds upon earlier results of the researchers, working together in some cases and separately in others, relating to various metric properties of Thompson's groups.This award supports US-Spanish cooperative research in mathematics. It provides travel support for Sean Cleary of The City College of New York and Jennifer Taback of the University at Albany to visit Jose Burillo of the Universitat Politecnia de Catalunya in Barcelona. This mathematical research lies in the area of group theory. A group is an abstract structure, which can often be visualized as a collection of symmetries of a geometrical object. Some common properties of a set of such symmetries can be made abstract in systematic ways. In this research, groups themselves are studied as geometric objects, and geometric insight can help to understand the algebraic structure of some groups more effectively. This research focuses on a family of groups known as Thompson's groups, after the researcher who first defined them. Thompson's groups can be understood geometrically using pairs of binary trees, an approach that relates this group to questions in theoretical computer science. The broader impacts of this project include these applications to algorithmic questions in computer science related to the shape of binary trees, international involvement and exchange between researchers interested in similar projects, and possible student involvement in research.

这个项目的目标是了解汤普森群的度量性质。 汤普森定义了一个有趣的群族，它出现在广泛的数学学科中，包括逻辑、同伦理论和离散群的测度理论。 汤普森群通常是最简单的，或者是唯一已知的一些奇怪的群论现象的例子。 尽管许多研究人员进行了大量的研究，但对这些群体的许多性质知之甚少。 本计画借由发展对这些群的凯莱图的更好了解，来研究汤普森群的度量结构。 这项研究建立在研究人员的早期成果的基础上，在某些情况下，在其他情况下，与汤普森集团的各种度量性质有关。 它为纽约城市学院的Sean Cleary和奥尔巴尼大学的Jennifer Taback访问巴塞罗那卡塔卢尼亚理工大学的Jose Burillo提供旅行支持。这一数学研究属于群论领域。群是一种抽象的结构，它通常可以被看作是几何对象的对称性的集合。一组这样的对称性的一些共同性质可以用系统的方法抽象出来。 本研究将群本身作为几何对象进行研究，几何洞察力有助于更有效地理解某些群的代数结构。这项研究的重点是一个家庭的群体被称为汤普森的群体，后的研究人员谁第一次定义他们。汤普森的群可以用二叉树对来几何地理解，这种方法将这个群与理论计算机科学中的问题联系起来。该项目更广泛的影响包括计算机科学中与二元树形状相关的算法问题的应用、对类似项目感兴趣的研究人员之间的国际参与和交流，以及学生可能参与研究。