# The Dixmier-Moeglin problem for D-varieties

## Rahim Moosa

### University of Waterloo

By a D-variety we mean, following Buium, an algebraic variety V over an algebraically closed field k equipped with a regular section s: V–> TV to the tangent bundle of V. (This is equivalent to the category of finite dimensional differential-algebraic varieties over the constants.) There are natural notions of D-rational map and D-subvariety. Motivated by problems in noncommutative algebra we are lead to ask under what conditions (V,s) has a maximum proper D-subvariety over k. (Model-theoretically this asks when the generic type is isolated.) A necessary condition is that (V,s) does not admit a nonconstant D-constant, that is, a D-rational map from (V,s) to the affine line equipped with the

zero section. When is this condition sufficient? I will discuss this rather open-ended problem, including some known cases.

Rahim Moosa received his doctorate in 2001 from the University of Illinois at Urbana-Champaign, with Anand Pillay as his advisor. After a series of postdoctoral positions, he joined the faculty at the University of Waterloo, where he is now Associate Professor. He studies model theory, especially in relation to differential algebra, fields, and number theory.