Central simple algebras are forms of matrix algebras, and their additional differential structures can be studied through categories of differential modules. In characteristic zero, ordinary derivations extend to forms under appropriate smoothness hypotheses, and Picard–Vessiot theory provides splitting fields that respect these structures. This talk reviews that background and then turns to iterative derivations, which retain higher-order information in positive characteristic. The categorical framework also lets us study iterative differential algebra structures on forms of finite-dimensional algebras with smooth automorphism schemes.

Let F be an iterative differential field with algebraically closed field of constants C, and let A be an iterative differential central simple F-algebra of degree n. The Picard–Vessiot field of the underlying iterative differential module splits A and is minimal among splitting extensions that introduce no new constants. Its iterative differential Galois group acts on the constant matrix algebra through a projective representation into PGL_n(C). Tannakian descent classifies the algebras of degree n split by a fixed Picard–Vessiot field in terms of projective representations of its Galois group. We show that these algebras arise as pushforwards of the Picard–Vessiot torsor Spec(T(K/F)) along suitable homomorphisms from the Galois group to PGL_n(C), and establish consequences of this description.

Using the central projection GL_n → PGL_n and Morita theory, we relate the subgroup properties PGL_n-cr, PGL_n-irr, and PGL_n-ind to complete reducibility, irreducibility, and indecomposability of the associated representation, and hence to right ideals of A stable under its iterative differential operators. We also discuss extension of iterative derivations: a prime-to-p exponent condition suffices for central simple algebras, but extension does not hold in general. This talk is based on joint work with Dr. Varadharaj Ravi Srinivasan.

I will report on joint work in progress with James Freitag and Omar León Sánchez. We construct a strongly minimal ordinary differential equation whose solution set remains strongly minimal after the differential field is expanded by an additional commuting derivation. From the viewpoint of DCF_{0,2}, this gives a strongly minimal definable set of infinite differential dimension. Some of my contributions to the project were developed with the assistance of AI, and I will comment on this aspect during the talk. Time permitting, I will also explain possible connections with the method of differential constraints in the theory of PDEs.