Kolchin Seminar in Differential Algebra 
 Hunter College 695 Park Ave. New York, NY 10021 General Telephone: 12127724000 
Academic year 20052006. Other years: 20022003 20032004 20042005 20062007 Current 
Sally Morrison, Bucknell University, Lewisburg, PA
New methods in the computation of differential idealsComputations in differential polynomial algebra inevitably lead one to consider ideals of the form [P] : M^{ }, where P is a finite set of differential polynomials and M is a multiplicative subset of the underlying differential polynomial ring. The work of Ritt and Kolchin exploits these ideals in the case that P is a coherent autoreduced set and M is the multiplicative subset generated by the initials and separants of the elements of P. More recent computational techniques depend on the use of more general sets P and M. In this talk we explore important properties of these more general ideals, and comment on the actual and potential use of these ideals in computational techniques.
Alexander Levin, Catholic University of America, Washington, DC
Differential dimension polynomialsThe role of Hilbert polynomials in commutative algebra and algebraic geometry is wellknown. Recall that the classical version of the existence theorem on Hilbert polynomial states that if R[X1, ... ,X n] is a naturally graded polynomial ring over a field K and M = q Mq is a finitely generated graded Rmodule, then each component Mq, considered as a vector Kspace, has finite dimension M(q) and for sufficiently large q, the function M(q) is a polynomial in q whose degree is at most n1. This polynomial is called the Hilbert polynomial of the module M. A similar role in differential algebra is played by differential dimension polynomials introduced by E. Kolchin in 1964.
In this talk we discuss basic facts about differential dimension polynomials, outline methods of computation of such polynomials, and consider their invariants. We also discuss the connection between differential dimension polynomials and the concept of strength of a system of differential equations introduced by A. Einstein in his last works on Relativity.
The talk does not assume any knowledge beyond the standard undergraduate abstract algebra course.
Lourdes Juan, Texas Tech University, Lubbock, TXSaturday, December 17, 2005
Generic PicardVessiot extensions for connectedbyfinite groupsLet G be a linear algebraic group over the algebraically closed field C. In this talk we will show how a generic PicardVessiot Gextension can be produced when the group G is the semidirect product of its connected component G^{o} by a finite group H, provided that the adjoint Haction on Lie(G^{o}) is faithful. The main ingredients are our previous construction of a generic extension with connected Galois group G, a characterization of Hequivariance based on the Clinearity of the adjoint action obtained in this work, and a criterion of MitschiSinger to produce a PicardVessiot extension with group H G^{o}.
Alexey Ovchinnikov, North Carolina State University, Raleigh, NC
Tannakian categories for parametric differential equationsTannaka's Theorem says that a linear algebraic group is determined by the category of vector spaces on which it acts. This has been used as a foundation of the PicardVessiot theory. We will discuss a similar result for linear differential algebraic groups and its connection with the Galois theory of parameterized linear differential equations.
Tobias Dyckerhoff, University of Pennsylvania, Philadelphia, PA
PicardVessiot theory over nonalgebraically closed fields of constantsDifferential Galois theory uses PicardVessiot extensions to describe the symmetries of linear differential equations. However, there are some difficulties when dealing with nonalgebraically closed fields of constants. Galois descent provides a nice method to understand this situation. I will present an overview of this approach and describe relations to the theory of Tannakian categories.
Jerald Kovacic, CCNY
Differential schemesAn affine differential scheme, X = diffspec R, is similar to an affine scheme, except that we start with a differential ring R and consider differential prime ideals. There is a canonical mapping of R into the ring of global section of X. In scheme theory this mapping is an isomorphism, not so for differential schemes. We can easily determine the kernel. It is the differential ideal of differential zeros. Surjectivity is missing because of the existence of differential units and the lack of a common denominator. We shall also discuss other "challenges", such as the existence of products. For differential group schemes we have the challenge that they need not be linear, R need not be a differential Hopf algebra. This is an elementary talk. We assume the audience knows the definition of spec but little else.
Saturday, March 18, 2006
Hans Schoutens, New York City College of Technology, CUNY
Russell Miller, Queens College, CUNY
Hrushovski's proof of the MordellLang Conjecture
Part I: Jetspaces (Hans Schoutens)
Part II: Modeltheoretic differential algebra (Russell Miller)The MordellLang Conjecture, stated below and presented in [1], does not involve any differential operators. Nevertheless, differential algebra plays a substantial role in Hrushovski's proof of this result, at least in the case of characteristic 0.
MordellLang Conjecture (relative version, for function fields): Let k_{0} K be two distinct algebraically closed fields. Let X be an infinite subvariety of an abelian variety A , both defined over K , and let Γ be a subgroup of A(K) of finite rank. Also suppose that X(K) Γ is Zariskidense in X(K) . Then either
(1) X(K) Γ is a finite union of cosets of subgroups of Γ , or(2) the data 'descend to k_{0} ', in the sense that there exist a subabelian variety B of A , an abelian variety B_{0} and a subvariety X_{0} of B_{0} both defined over k_{0} , and a bijective morphism h from B onto B_{0} , such that X = a + h^{1}(X_{0}) for some aA .Part I (jetspaces):
After Hrushovski's original proof was published, Pillay and Ziegler [3] found a way around part of the heavyduty theory (Zariski geometries) that was needed in the proof by introducing the differential analogue of a jetspace. Modeltheory is now only present in the background of this part of the proof.
Part II (differential algebra):
We introduce the model theory of the situation and investigate the reasons why derivations are useful in this context, and what one can do with differential algebra that could not have been done (at least, not so easily) with ordinary algebraic geometry.
References:
[1] Bouscaren, E., ed., Model Theory and Algebraic Geometry: An introduction to E. Hrushovski's proof of the geometric MordellLang conjecture. (Berlin: Springer, 1999.)
[2] Marker, D., Model Theory of Differential Fields. (Berlin: Springer, 1996.)
[3] Pillay, A. and Ziegler, M., Jet spaces of varieties over differential and difference fields. Selecta Math. 9 (2003), no. 4, 579599.
Oleg Golubitsky, School of Computing, Queen's University, Kingston, OntarioSaturday, May 6, 2006
A bound for the orders of derivatives in the RosenfeldGröbner algorithm
(in collaboration with M. Kondratieva, M. Moreno Maza, and A. Ovchinnikov)We consider the RosenfeldGröbner algorithm for computing a regular decomposition of a radical differential ideal generated by a set of ordinary differential polynomials in n indeterminates. For a set of ordinary differential polynomials F , let M(F) be the sum of maximal orders of differential indeterminates occurring in F . We propose a modification of the RosenfeldGröbner algorithm, in which for every intermediate polynomial system F , the bound M(F) ≤ (n1)!M(F_{0}) holds, where F_{0} is the initial set of generators of the radical differential ideal.
In particular, the resulting regular systems satisfy the bound. Since regular ideals can be decomposed into characterizable components algebraically, the bound also holds for the orders of derivatives occurring in the characteristic decomposition of a radical differential ideal.
1:30  2:30 PM
XiaoShan Gao, Institute of Systems Sciences, Chinese Academy of Sciences
Difference characteristic set and resolventIn this talk, we will give an introduction to our current work on computational difference algebra via the characteristic set method. We first prove several basic properties for difference ascending chains including a necessary and sufficient condition for an ascending chain to be the characteristic set of its saturation ideal and a necessary and sufficient condition for an ascending chain to be the characteristic set of a reflexive prime ideal. We then propose an algorithm to decompose the zero set of a finite set of difference polynomials into the union of zero sets of certain ascending chains. As a consequence of the zero decomposition, we give a new algorithm for the perfect ideal membership problem. Finally, we introduce a new theory of resolvent systems for certain difference ideals and introduce algorithms to compute them.
2:50  3:50 PM
Lourdes Juan, Texas Tech University, Lubbock
On the structure of PicardVessiot extensions
(joint work with Arne Ledet)Let k be a differential field of characteristic zero with algebraically closed field of constants. A PicardVessiot extension K k is the function field of a k irreducible G torsor, where G is the differential Galois group of the extension. In this talk we will show how to construct examples of extensions which are the function field of nontrivial G torsors and that, when a good description of the torsors is available, generic extensions for G can be produced.
For lecture notes click here , see also the web site
4:10  5:10 PM
Julia Hartmann, University Of Pennsylvania, Philadelphia
Inverse problems in differential Galois theory
5:30  6:30 PM
Yang Zhang, Brandon University, Brandon, Canada
Noncommutative existence and uniqueness theory for partial differential equations
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