Finite and multiplicative algebraic group actions on affine spaces.
Peter Russel
(Mc Gill University - Canada)

Consider an algebraic action of a reductive group G on affine space A^n (in characteristic 0). Is the action linear in suitable coordinates for A^n? This is not always the case if n>3. I will discuss the positive answer for n=3 and G the multiplicative group G= G_ m, and for a finite group G and n=3 and for G=G_m and n=4, under the condition that G fixes a variable. Already with this very restrictive condition these cases appear to be quite challenging.