Algebraic Groups and Graph Colouring

代数群和图形着色

• 批准号: RGPIN-2017-05074
• 负责人: Wehlau, David
• 金额: $ 1.75万
• 依托单位: Royal Military College of Canada
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=679538
• 关键词: Algebraic; Groups; Graph; Colouring

I pursue research in (at least) two separate areas of mathematics.******Invariant theory is concerned with the study of symmetry. The collection of symmetries that an object possesses can be collected together into a mathematical object known as a group. Properties of the object can be deduced from the structure of its group of symmetries. Invariant theory has many modern applications, including applications to computer vision, satellite and outer space navigation, fingerprint identification and in materials sciences. ******Invariants were originally introduced in order to distinguish objects from one another. One aspect of invariant theory which I have been actively studying is the topic of separating invariants. The goal in this field is to find small numbers of invariants which serve just as well as the full set of invariants to distinguish objects from one another. For example, the problem of finding a small collection of measurable properties of a fingerprint which allows a computer to determine whether two fingerprints are from the same finger or not. ******I also study discrete mathematics. This includes studying ways to count things, studying graphs, and studying some aspects of logic. I am interested in bounding the chromatic number of graphs. This has very many practical applications. Perhaps the easiest to describe is the problem of scheduling. Given a number of processes competing for some scarce resource, how should we schedule access to this resource to optimize whatever process is running. As a concrete example: given a list of courses and students registered in various of these courses, how many classrooms are required to allow each student to attend all of his classes. This is known to be an extremely difficult problem which cannot be solved efficiently.******This research provides an excellent avenue for the education of students and junior researchers. I will supervise 9 such people per year including 3 undergraduate students, 4 graduate students and 2 postdoctoral fellows. These people will learn new areas of modern mathematics and will develop reasoning and communication skills that will serve them well in whatever career they pursue.*****************

我从事（至少）两个独立的数学领域的研究。不变量理论是关于对称性的研究。一个对象所具有的对称性的集合可以被集合在一起成为一个数学对象，称为群。物体的性质可以从它的对称群的结构中推导出来。 不变量理论有许多现代应用，包括计算机视觉，卫星和外层空间导航，指纹识别和材料科学的应用。 * 不变量最初是为了区分对象而引入的。 我一直在积极研究的不变量理论的一个方面是分离不变量的主题。 这个领域的目标是找到少量的不变量，这些不变量与完整的不变量集一样，可以将对象彼此区分开来。 例如，找到指纹的可测量属性的小集合的问题，该集合允许计算机确定两个指纹是否来自同一手指。 ** 这包括学习计数的方法，学习图形，学习逻辑的某些方面。 我感兴趣的是图的色数的界。 这有很多实际应用。 也许最容易描述的是调度问题。 给定多个进程竞争某个稀缺资源，我们应该如何调度对该资源的访问，以优化正在运行的进程。 举一个具体的例子：如果有一份课程表，而学生已登记修读这些课程，请问每名学生需要多少间课室才可修读所有课程？ 这是一个非常困难的问题，无法有效解决 *。本研究为学生和初级研究人员的教育提供了一个很好的途径。 我每年将指导9名这样的人，包括3名本科生，4名研究生和2名博士后。 这些人将学习现代数学的新领域，并将发展推理和沟通技能，这将有助于他们在任何职业生涯中，他们追求。