Let G be an algebraic group and V is a (rational) representation of G. Then G also acts on the coordinate ring K[V] (K is some field). The polynomial functions on V which are invariant under G form a subring, the invariant ring K[V]G. These rings where already studied (although not in this language) in the 19th century, especially where G is SL2 or an other classical group. With David Hilbert, an new era for invariant theory arose (and another era ended). His papers of 1890 and 1893 not only proved one of the major results in invariant theory, but also build the foundation of what is now commutative algebra and algebraic geometry.
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 14th problem (one of the problems he proposed at the beginning of the 20th century to keep everyone busy). Before, in 1890, he proved that for G=SLn, G=GLn 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 14th problem where G is not reductive.
I am very much interested in the constructive aspect of invariant theory. One of my contributions is an algorithm for computing a finite set of generating invariants for K[V]G if G is linearly reductive. Let b(V) be the smallest integer d such that K[V]G is generated by invariants of degree less or equal than d. I also obtained some good upper bounds for b(V).
A quiver Q is just a graph. If we attach to each vertex a vector space and to each arrow a linear map, then this is called a representation of the quiver Q. Quiver representations give a natural generalization of some linear algebra problems.
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 particularly interested in the connection between quiver representations and the representation theory of GLn. Studying the behaviour of ``generic'' representations of quivers, one can obtain results about Littlewood-Richardson coefficients.
Automorphisms of Affine space
The algebraic automorphism group of Kn (K an algebraically closed field, for example the complex numbers) is very complicated. There are still many open problems relating to this automorphism group. For example, the Zariski Cancellation conjecture: Conjecture: If XxK is isomorphic to Kn+1 for some affine variety X, then X must be isomorphic to Kn.
or this conjecture by Abhyankar and Sathaye:
Conjecture: If f:Kn--->K is a polynomial such that f-1(0) is isomorphic to Kn-1, then any fiber f-1(a) is isomorphic to Kn-1 (or even stronger: f must be a polynomial coordinate).
and then there is the linearization conjecture: 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.