Algebraic Groups and Graph Colouring

我从事(至少)两个不同数学领域的研究。*不变理论关注对称性的研究。一个对象所具有的对称性的集合可以聚集在一起形成一个称为群的数学对象。物体的性质可以从它的对称组的结构中推断出来。不变量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。*最初引入不变量是为了将对象彼此区分开来。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，这些不变量的作用与完整的不变量集一样，以区分对象和其他对象。例如，寻找指纹的可测量属性的小集合的问题，允许计算机确定两个指纹是否来自同一手指。*我也学习离散数学。这包括学习计算事物的方法，研究图表，以及研究逻辑的某些方面。我对图的色数的界很感兴趣。这具有非常多的实际应用。也许最容易描述的是日程安排问题。假设有多个进程竞争一些稀缺资源，我们应该如何安排对该资源的访问以优化正在运行的任何进程。举一个具体的例子：给出一份课程清单和注册这些课程的学生，需要多少个教室才能让每个学生参加他的所有课程。这是一个非常困难的问题，无法有效地解决。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，其中本科生3人，研究生4人，博士后2人。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，这将为他们在任何他们所追求的职业中提供良好的服务。

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