On algebraic geometry over division rings

We shall survey recent developments concerned with foundational aspects of quaternionic algebraic geometry:

1. A quaternionic Nullstellensatz for the ring R of polynomials in n central variables over the quaternion algebra H, in both abstract form (due to the author and Alon) and explicit form (due to M. Aryapoor).
2. A theorem about the geometry of zero sets of polynomials in R: If a polynomial vanishes on all common zeros with commuting coordinates of a left ideal J in R , then it vanishes on all common zeros of J in H^n. This result confirmed a conjecture of Gori, Sarfatti and Vlacci.
3. Study of contraction properties of one-sided ideals in polynomial rings over division rings. In particular, we show that if M is a maximal left ideal in the polynomial ring D[x], where D is a division ring, then the contraction of M to D need not be maximal. This resolved a question of Amitsur and Small from 1978. We shall discuss connections between this question to recent works and the implications to algebraic geometric over division rings.
4. A Nullstellensatz for quaternionic polynomial functions, a generalization to arbitrary centrally finite division rings by Bao and Reichstein, and an extension of the Ax-Grothendieck theorem to polynomial functions over centrally finite division rings.

Joint works with Gil Alon, Adam Chapman.