Type: Seminar

Distribution: World

Expiry: 21 Feb 2008

Speaker: Peter Fleischmann (Kent) Title: Some Aspects of Modular Invariant Theory of Finite Groups Time & Place: 2:30-3:30pm, Thursday 21 February, Carslaw 535. Abstract: Let $k$ be a field and $G$ a finite group, acting on the polynomial ring $A:=k[x_1,\dots,x_d]$ by graded $k$-algebra automorphisms. The ring of invariants $A^G:=\{f\in A\ |\ g(f)=f\}$ is the main object of study in Invariant Theory. The theory is very well developed in the ``classical case", where the characteristic of $k$ is zero, but far less so in the case of positive characteristic, in particular the ``modular case", where the characteristic divides the group order $|G|$. In that case there are open questions about the constructive complexity of $A^G$, measured by degree bounds for generators, and about the structural complexity, measured by the depth (=length of maximal regular sequence, or ``cohomological co-dimension) of $A^G$ as a module over a homogeneous system of parameters. In my talk I will, after a brief introduction, report on some recent results dealing with both types of questions. -------------------------------------------------------------------------------- Speaker: Michael Pohst (Berlin) Title: On Solving Diophantine Equations over Global Fields Time & Place: 4:00-5:00pm, Thursday 21 February, Carslaw 535. Abstract: We present methods for the resolution of decomposable form equations over global fields. In general, those equations are reduced to unit equations. Algorithms for solving the latter differ substantially in the number and function field case. Thue and norm form equations will be discussed in greater detail. Also, the fastest known method for computing all integral points on Mordell curves y^2=x^3+k will be presented.