- Main topic: Linear algebraic groups and their associated algebraic structures, over arbitrary fields
- Other key topics: Essential dimension, cohomological invariants, motivic decompositions of projective homogeneous varieties, and G-torsors.
- Some matrix groups (that is, a set of invertible matrices such that the product of any two elements is again in the set) are defined by polynomial equations, for example SLn, the group of matrices of determinant one. Such objects are called linear algebraic groups (See also: What is a linear algebraic group?). Over the complex numbers, the simple linear algebraic groups have an elegant and easy to understand classification, given by their ''root data''. This classification was originally discovered by Wilhelm Killing in the late 1800's in what has been refered to as "the greatest mathematical paper of all time". When one changes the base field to something other than the complex numbers, the classification still holds (at least for split groups), and this provides a deep link between the discrete world (for example, when the base field is finite one obtains finite groups of Lie type), and the continuous world (for example, when the base field is real or complex one obtains the Lie groups).
- I am interested in taking a PhD student. Such a student would be expected to have some familiarity with algebraic geometry. If you are interested, you can email me.