Online Workshop in Memory of Ray Hoobler

In algebraic geometry, a fruitful approach to study algebraic varieties (spaces that are locally defined by systems of polynomial equations) is through the machinery of Grothendieck topologies. This is a way to enlarge the category of open subsets of the space (of which there are too few in a purely algebraic context), with the hope that the resulting sheaf cohomology will yield useful algebraic invariants of the variety (for example, the genus of a Riemann surface can be obtained in such a way).

In the last several years, Ray Hoobler had been working on a program aimed at developing analogous machinery to study differential algebraic varieties (spaces that are locally defined by systems of polynomial differential equations). This talk will begin with a brief and informal introduction to Grothendieck topologies aimed at providing the non-expert with a reasonable sense of what they are and how they have been useful in the past. I will then mention some of the technical difficulties that arise in the development of similar technology in the differential context, and how to overcome them. I will conclude with a description of the Picard-Vessiot topology for differential varieties (developed jointly with Ray Hoobler) and explain why it is a reasonable differential analogue of the classical étale topology for algebraic varieties.

The talk will give a brief introduction to patching and its recent applications in differential Galois theory, and then suggest a new use of patching in differential algebra.

This is a report on recent joint work with L.E.Miller on the arithmetic PDE theory of elliptic curves and modular curves. The role of partial differentiation operators is played by the Fermat quotient operators attached to generators of the absolute Galois group of a local field. Diophantine applications will be explained, including a reciprocity law for the arithmetic Manin maps of 2 distinct elliptic curves.

The differential Brauer monoid of a differential commutative ring R will be defined. Its elements are the isomorphism classes of differential Azumaya R algebras with operation from tensor product subject to the relation that two such algebras are equivalent if matrix algebras over them are differentially isomorphic. The Bauer monoid, which is a group, is the same thing without the differential requirement. The Brauer monoid of the ring of constants injects into the differential Brauer monoid of R with image the submonoid of invertible elements. Some examples will be presented.