On the occasion of Gerhard Pfister's 60th birthday, the conference "Algebraic Geometry and Computer Algebra" will be held at the University of Kaiserslautern from May 31 to June 2, 2007.

## Tentative Schedule:

### All talks will be held in Building 42, Room 110.

### Thursday

**14:00 - 14:30**- Opening Cermony
**14:30 - 15:30**- W. Decker: Syzygies in Algebraic Geometry
I will, mainly through examples, discuss the use made of syzygies in computational experiments related to geometric problems.

**15:45 - 16:45**- T. Mora: Combinatorial duality approach to 0-Dimensional ideal
A survey on combinatorial duality approach: Cerlinaco-Mureddu Algorithm, the Axis-of-Evil Theorem, Macaulay's Algorithm, Noether Decomposition

**17:00 - 18:00**- Le D.T.: Compactification of polynomial functions in two variables
In this lecture we shall indicate how to compactify polynomial functions in two variables. We shall consider the particular case of rational polynomial in two variables and show that complex simple rational polynomials belong to a Jacobian pair if and only if it is a coordinate after composing with an automorphism of the plane.

### Friday

**09:00 - 10:00**- B. Kreussler: Bundles on Elliptic Curves and Yang-Baxter Equations
What is now called the Yang-Baxter equation was introduced independently by Yang (1967) who considered a quantum mechanical many body problem and by Baxter (1972) in statistical mechanics. In 1982, Belavin and Drinfeld classified solutions of the classical Yang-Baxter equation. They showed that solutions are either elliptic, trigonometric or rational. In 2002, Polishchuk found a surprising relationship between structural properties of derived categories and solutions of the associative Yang-Baxter equation. In joined with Igor Burban we introduce a relative version of Polishchuck's construction and use it to study degenerations of solutions of the classical Yang-Baxter equation. This is related to degenerations of elliptic functions.

**10:30 - 11:30**- A. Campillo: Singularities and Poincaré series
In join work with F.Delgado and S.Gussein-Zade Poincare series for rather general multiindex filtrations of local rings had been introduced and developed. For many kinds of singularities, appropriated Poincare series for them are shown to be directly related to their geometry. Several authors have shown that this happens for classes of singularities which include the following ones: plane curves, rational of surfaces, equivariant two dimensional, normal surfaces, quasihomogeneous complete intersections, affine toric, and quasiordinary hypersurfaces. We show how all those cases can be unified in a general algebraic geometric context. For it, we consider Poincare series for filtrations which are given by sets of valuations which depend only on the local ring as invariants for the singularities. We show how such invariants behaves and allow, in particular, to recover the main features of each one of the classes.

**11:45 - 12:45**- I. Luengo: Weighted-Iomdine surface singularities
I will report on a joint work with E. Artal, J. Fernandez de Bobadilla and A. Melle, related with Zariski's multiplicity question. We will consider Weighted-Iomdine singularities defined in $(\bc^3,0)$ by a series $g:=g_d+g_{d+k}+\dots$, where $g_m$ is weighted-homogeneous such that the singularities of $g_d$ in the weighted-projective plane do not intersect the curve defined by $g_{d+k}$. We will give a formula for the Milnor number and relate it with the geometry and resolution of the singularity to give interesting examples of such singularities. We give a negative answer to Question B of Zariski for surface singularities.

**12:45 - 14:30**- Lunch break
Enjoy your lunch and/or have fruitful discussions.

**14:30 - 15:30**- A. Laudal: 5.11 and the impact on moduli theory
Working on the moduli space of isolated plane curve singularities in the 1980'ties, Gerhard Pfister and I needed some good examples. One such was the miniversal family of the singularity x^11+y^5. Most of the calculations were done by Gerhard, and these probably lead Gerhard into the field of computer algebra, which later gobbled him up. This, I assume, has been a blessing for himself and for many others. To me it meant a complete turn-around, and I turned non-commutative. This talk is going to explain why, and give some glimpses of the impact 5.11 had on my understanding of moduli theory.

**16:30 - 17:30**- P. Schenzel: On "small" non-Cohen-Macaulay factorial domains
The first aim of this talk is the construction of unique factorization domains in small dimension. We construct a unique factorization domain of dimension 3 which is not a Cohen-Macaulay ring. Moreover there is an example of a four-dimensional affine ring S over a field k with the property that S is a non-Cohen-Macaulay unique factorization domain whenever Char k = 2, while it is a Gorenstein non-factorial ring for Char k != 2. The arguments for the proofs are conceptional as well as based on the Computer Algebra System Singular.

### Saturday

**09:15 - 10:15**- D. Popescu: Artin approximation, general Neron desingularization and applications
This is a small survey old and new on Artin approximation and connected problems.

**10:30 - 11:30**- H. Kurke: Formal ribbons and higher dimensional Kricever Correspondence
The " classical" Kricever correspondence relates 1-dim geometric data (Curve with a choice of a point, a formal parameter in this point, a torsion free coherent sheaf and a formal trivialization in the point) with certain points in an infinite dim Grassmannian of a locally linearly compact vectorspace V. It relates the KP eqation or rather KP hierarchy with deforming the geometric data resp with flows on the Grassmannian. I will report an recent progress in higherdim. Situation (joint work with D.Osipov and A. Zheglov) The KP Hierarchy is to be replaced by a System of Differential equations introduced by Parshin and modified by Zheglov, the geometric data are replaced by "formal ribbons", which include as spacial case surfaces with a distinguished ample curve and a point on the curve plus formal parameters related to the curve and the point. There is also an analogue of the Sato Grassmannian.