**The Kolchin Seminar in Differential Algebra**

**Saturday March 15 ^{th} 2003**

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**Jean-Pierre Ramis**

Université
Paul-Sabatier (Toulouse) and Institut Universitaire de France
**Invariants of q-Difference Equations in the Complex Domain and
Galois Theory**

In these two lectures, we will present a complete theory of (algebraic and transcendental) invariants of linear

"

This program was completed quite recently by J. Sauloy, C. Zhang and J.P. Ramis.

**Part 1: The Fuchsian Case **

2:00 - 3:00 PM

**Abstract: ** We will begin with the Fuchsian (or regular singular) case. We
will first recall briefly the invariant theory for the
differential regular singular case (Riemann-Hilbert theory). After
we will develop the parallel *q*-difference case. The classical
monodromy representation is more or less replaced by a Birkhoff
connection matrix. In our new approach this matrix is elliptic.
If we perform a continuous limit (*q* -> 1),
*q*-difference equations tend to differential equations and there
is a beautiful confluence relation between the discrete and the
continuous theories of invariants.
In the linear complex differential regular case there is a nice
relation between the invariant theory and the differential Galois
theory: "the" differential Galois group "is" the Zariski closure
of the image of the monodromy representation. We will present a
similar result in the *q*-difference case. The first step (in the
regular case) is due to P. Etingof. The general regular singular
case is a lot more delicate; a clear presentation needs the use of
groupoids and flat fiber spaces on elliptic curves.
Finally, for the regular singular case, the situation is quite
nice and we have more or less a complete theory: invariants,
confluence, Galois theory, and their relations. Moreover for the
important family of *q*-analogues of hypergeometric equations (basic
hypergeometric equations), it is possible to compute explicitly
everything.
In the irregular case, even if we made big progresses, we did not
reached the same degree of achievement...

**Evgueny V. Pankratiev**

Moscow State University
**Some Approaches to the Construction of Differential Gröbner Bases**

3:30 - 4:30 PM

**Abstract:**
The notion of Gröbner bases introduced by B. Buchberger plays a key role
in computer algebra. Along with Gröbner bases, other standard bases
are used in constructive theory of polynomial ideals, e.g.,
involutive bases and characteristic sets.
These notions can also be used in differential modules, i.e., in (left)
submodules of finitely generated (left) free modules over the ring of
differential operators.
As for differential ideals in rings of differential polynomials,
there are several nonequivalent approaches to generalizing the
notion of Gröbner bases. Unfortunately, they are not so ``good'' as the
polynomial Gröbner bases: they can be used only in some classes of
differential ideals and their properties are much weaker than those of
Gröbner bases.
There are much more problems than solutions.
Some results in this area obtained by the ``Moscow team'' will be presented.

**Jean-Pierre Ramis**

Université
Paul-Sabatier (Toulouse) and Institut Universitaire de France
**Invariants of q-Difference Equations in the Complex Domain and
Galois Theory
Part 2: The Irregular Case **

5:00 - 6:00 PM

**Abstract:** We will first recall briefly the invariant theory in the irregular
differential case. The divergence of formal power series solutions
play a central role. There exists a delicate theory of
"canonical" resummations of such series and the ambiguities in
the resummations give rise to the Stokes phenomena which give new
analytic invariants (a new sort of monodromy). Summability theory
is strongly related to some Gevrey estimates. In the *q*-difference case the situation is in some sense
less and more difficult. Less difficult because surprisingly there exists locally at zero
and infinity a canonical *analytic* filtration (related to a
Newton polygon). In the differential case this filtration is only
*formal*. More difficult because the *q*-analogue of *k*-summability is quite
delicate. An important fact is that it does not exist a natural
process for choosing a *q*-analogue. In particular, if one wants a
*q*-analogue of the Laplace transform (which is needed for
*k*-summability theory), there is some choices:
-- Kernel choices: we must choose among three quite natural
*q*-analogue of the exponential function.
-- Contour choices: we must choose a continuous contour
(continuous spiral) or a discrete contour (discrete spiral).
These choices are strongly related to a choice of *q*-constant
fields. We developed systematically the summation theory in all
these directions, but our aim was to get at the end elliptic
matrices as resummations ambiguities (*q*-Stokes phenomena). We
succeeded using good choices for the kernels (theta functions) and
the contours (discrete spirals). At the end we built a complete
theory of invariants along these lines. We will present shortly
three aspects: algebraic, geometric and analytic (*q*-Gevrey
estimates and *q*-analogue of multisummability). This finishes
Birkhoff program.
Now, what about continuous limit (confluence) and Galois theory?
We just began the work in these directions. We can guess on some
examples that there will be a limit process from the *q*-Stokes
phenomena towards the Stokes phenomena. But we have only a
conjecture (involving cohomology spaces).
The relation between the invariant theory and the Galois theory
remains delicate in the irregular case. The main difficulty of the
Fuchsian case (in general the natural fiber functors are not
compatible with tensor products) is even more serious now. This is
strongly related with classification of fiber bundles on elliptic
curves and geometric non abelian class field theory...

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**Coffee and tea will be served beginning at 1:30 PM
Room HW706**

**Click here for directions to Hunter College
and location of room**