One of the major problems is whether K[V]^{G} is always
finitely generated as an algebra over K. This is known
as Hilbert's 14^{th} problem (one of the problems he proposed
at the beginning of the 20^{th} century to keep
everyone busy). Before, in 1890, he proved that for G=SL_{n},
G=GL_{n} and other classical groups. It can be shown
that K[V]^{G} is finitely generated for any reductive
group. However, Nagata found a counterexample for Hilbert's 14^{th}
problem where G is not reductive.

I am interested in the constructive aspect of Invariant Theory. I have worked on algorithms for finding generators of invariant rings, and finding upper bounds for the degrees of a minimal system of generators. Together with Gregor Kemper I wrote a book on Computational Invariant Theory. More recently, I have studied Invariant Theory in relation to Complexity Theory, and the Graph Isomorphism Problem. Currently I am studying tensor invariants (a classical topic in Invariant Theory) in relation to wheeled PROPs (a structure recently introduced by Markl, Merkulov and Shadrin).

There are several reasons for studying quiver representations. To each quiver one can associate its path algebra. Any finite dimensional associative algebra is just a path algebra modulo some ideal (up to Morita equivalence). So study of finite dimensional associative algebras leads to the study of quivers. Results of Kac, Lusztig and Nakajima also link quiver representations to Kac-Moody algebras and Quantum groups etc.

I am interested in the connection between quiver representations
and the representation theory of GL_{n}. Studying the behaviour
of ``generic'' representations of quivers, one can obtain results
about Littlewood-Richardson coefficients (tensor product multiplicities that appear
in the representation theory of the general linear group).
Quivers with (super)potentials appear in String Theory, but also in my joint work
with Weyman and Zelevinsky on Cluster Algebras.

**Conjecture:** If XxK is
isomorphic to K^{n+1} for some affine variety X,
then X must be isomorphic to K^{n}.

or this conjecture by Abhyankar and Sathaye:

**Conjecture:** If f:K^{n}--->K is a polynomial
such that f^{-1}(0) is isomorphic to K^{n-1},
then any fiber f^{-1}(a) is isomorphic to
K^{n-1} (or even stronger: f must be a polynomial
coordinate).

and then there is the linearization conjecture:

**Conjecture:** Let G be a linear algebraic group
acting rationally on K^{n}. Is the action of
G linearizable, i.e., is the action linear after a polynomial
change of coordinates?

To this last conjecture there are counterexamples in positive characteristic (Asanuma), if G is not commutative (Schwarz, Knop and others), and in the holomorphic category (Kutzschebauch and myself). However it is still open if G is abelian and K has characteristic 0.