## Differential Schemes and Differential Cohomology

- Start Date: 06/25/2012
- End Date: 06/26/2012

Robin Cockett

*Title: Tangent Structure and Differentials Categorically *

Dingkang Wang and Yao Sun

*Title: A Signature-Based Algorithm for Computing Bases in Solvable Polynomial Algebras *

Carlos Arreche

*Title: Solving Linear Differential Equations with Parameters: an algorithmic approach with applications to arithmetic geometry *

University of Calgary

Please note that the workshop is an extension of the workshop of the same name taking place at BIRS on June 22-24, 2012.

The Calgary Workshop will be based in part on what transpires in Banff, with some participants to present further details on topics discussed at the Banff portion of the Workshop. These individuals will be selected at the Banff Workshop.

**SCHEDULE OF TALKS**

More talks than appear below are expected, but several of these will be determined at a BIRS workshop in Banff to be held the preceding weekend. Those interested in speaking should contact the organizers, Richard Churchill and Yang Zhang.

__Monday, 25 June__

__9:00-10:00__

Tangent Structure and Differentials Categorically

*Robin Cockett, University of Calgary*

In 1984 Jiri Rosicky proposed in a short paper an abstract definition for a tangent bundle functor which generalized the usual tangent bundle functor of differential geometry. From this description Rosicky was able to show that one can, quite abstractly, define the Lie bracket on vector fields. Furthermore, when the tangent structure is “representable”, he exhibited a connection to synthetic differential geometry.

More recently, in 2009, Rick Blute, Robert Seely, and the speaker introduced the notion of a (Cartesian) differential category to provide an abstract description of differentiation. This was motivated not only by the desire to capture the essential algebraic manipulation underlying ordinary “multivariable” differential calculus but also by structures arising from Computer Science, Linear Logic, and combinatorics.

A natural question to ask concerns the relationship between these notions. Slightly generalizing Rosicky’s ideas to “tangent structure” allows one to link these notions by an adjoint. Every (Cartesian) differential category has tangent structure. Furthermore, within a category with (Cartesian) tangent structure there is always a full subcategory of “differential objects” – they are the tangent spaces – which, as their name suggests, form a differential category. Sometimes, as is the case with differential manifolds, one can actually “generate” the larger category from its differential objects.

Of interest are the constructions to which categories with tangent structure are closed. Two constructions of interest are forming presheave and scheme categories. The former has been well-investigated and it is known that the differential structure does not (quite) transfer. The latter case has not been as well investigated, however, once again, the structure does not in general transfer – although there are special cases in which it does.

__10:00-10:30__

A general discussion, on his talk, led by Robin.

__10:30-11:30__

A Signature-Based Algorithm for Computing Bases in Solvable Polynomial Algebras

*Dingkang Wang and Yan Sun - University of Manitoba *

*(Wang) and Key Laboratory of Mathematics Mechanization, Academy of Mathematics and Systems Science (Sun) (joint work with Yang Zhang)*

Signature-based algorithms, including F5, F5C, G2V and GVW, are efficient algorithms for computing Groebner bases in commutative polynomial rings. In this paper we present a signature-based algorithm to compute Groebner bases in solvable polynomial algebras which include the usual polynomial rings and some non-commutative polynomial rings such as the Weyl algebra. The generalized rewritten criterion (proposed in Sun and Wang 2011) is used to construct this new algorithm. When this new algorithm uses the order implied by GVW, its termination is proved without special assumptions on the computing order of critical pairs. Data structures similar to F5 can be used to speed up this new algorithm, and Groebner bases for corresponding syzygy modules can be obtained from the outputs in polynomial time. Experimental data shows that most redundant computations can be avoided in this new algorithm.

__11:30-12:00__

A general discussion, of this talk, led by the speakers

**12:00-13:30**

Lunch

__13:30-14:30__

Solving Linear Differential Equations with Parameters: An Algorithmic

Approach with Applications to Arithmetic Geometry

*Carlos Arreche, Graduate Center, City University of New York*

Towards the end of the last century J. Kovacic constructed an algo-rithm which: (i) determines the differential Galois group of a second- order linear homogeneous ordinary differential equation with coefficients in a field of rational functions over an algebraically closed field of characteristic 0; and (ii) determines from that group if the equation admits “elementary” solutions. We have been successful in extending Kovacic’s methods to cover the case of one-parameter families of second-order linear homogeneous equations, and the talk will outline the basic ideas.

The Galois theory of differential equations with parameters is originally due to Peter Landesman, and the special case of linear differential equations with parameters (Parametrized Picard-Vessiot Theory) has

been recast quite recently in close analogy with the classical Picard- Vessiot theory by P. Cassidy and M. Singer. In recent work of Gorchin- skiy and Ovchinnikov the work of Cassidy-Singer has been related to Gauss-Manin connections. We will derive the Picard-Fuchs equation as a simple application of our algorithm in order to illustrate our methods, and discuss other potential applications to arithmetic and algebraic geometry if time permits.

__14:30-15:00__

A general discussion, of his talk, led by Carlos.

__Tuesday, 26 June
__

**10:30-11:30**

Quasi-Prime Ideals*Bill Keigher, Rutgers-Newark*

__ 11:30-12:30__Tangent Structure and Differentials Categorically - Part II

*Robin Cockett, University of Calgary *

**ALL REMAINING TALKS FOR MONDAY AND TUESDAY ARE TO BE ANNOUNCED.
**

Please contact Rick Churchill for further information.