Grant: Discovery Grants Program - Individual

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