Algebraic Groups and Galois Geometries

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I pursue research in 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. One of the earliest sources of invariant theory was attempts to understand perspective. Invariant theory has many modern applications, including applications to computer vision, satellite and outer space navigation, fingerprint identificaton and materials sciences. I also study Galois geometries. This is the study of geometry in spaces which contain only finitely many points. I am particularly concerned with caps. These are sets of points for which no three lie on the same line. Caps are equivalent to certain codes. These codes are used to encode information compactly and in a form which minimizes the errors in transmission. Indeed these compact forms provide a method for correcting any transmission errors that might occur. Invariant theory also has applications to coding theory.