Algebraic Groups and Graph Colouring

我从事（至少）两个独立的数学领域的研究。******不变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成一个数学对象，称为群。物体的性质可以从它的对称群的结构中推导出来。不变性理论有许多现代应用，包括在计算机视觉、卫星和外层空间导航、指纹识别和材料科学中的应用。******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问题。这个领域的目标是找到少量的不变量，这些不变量的作用和完整的不变量集一样好，可以区分对象。例如，找到指纹的一小部分可测量属性的问题，使计算机能够确定两个指纹是否来自同一根手指。******我也学习离散数学。这包括学习数数的方法，学习图表，以及学习逻辑的某些方面。我对图的色数边界很感兴趣。这有很多实际应用。也许最容易描述的是日程安排问题。给定许多进程竞争某些稀缺资源，我们应该如何安排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：给定一个课程列表和在这些课程中注册的学生，需要多少教室才能允许每个学生参加所有的课程。众所周知，这是一个很难有效解决的问题。******这项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，其中本科生3名，研究生4名，博士后2名。这些人将学习现代数学的新领域，并将发展推理和沟通技巧，这将在他们从事的任何职业中很好地服务于他们。*****************

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.*****************