Transformation groups appear in many areas of mathematics--in topology, geometry, algebra, number theory--and quite often play a fundamental role. They help to understand the symmetries of an object and its intrinsic properties, to distinguish between different objects, and to construct new ones, e.g. via orbit spaces or universal coverings.
In the algebraic setting, the algebraic transformation groups play an important role in almost all classification problems, via the so-called moduli schemes. One wants to describe or classify the orbits under the action of an algebraic group. There is a major obstacle in this approach since--in general--there exist non-closed orbits, and so we cannot expect that the orbit space has a reasonable structure. There are several ways to get around this problem. One is to look at the algebraic quotient where one identifies two points in case their orbit closure have a non-trivial intersection. The result is a nice variety, at least in the case of a reductive group action on an affine variety. Another possibility is via stability considerations where we remove the unstable points in order to get an open dense set where all orbits are closed.
The situation is much more complicated if the acting group is not reductive. The first interesting case is the study of additive group actions, i.e. actions of the additive group $K+$ or, equivalently, of locally nilpotent vector fields. Already in this case many things can go wrong. E.g. the ring of invariants might not be finitely generated (Hilbert's 14th Problem). On the other hand, the existence of many different additive group actions could be an indication that the underlying variety is rational or at least rationally connected. The famous Makar-Limanov invariant is a measure for the existence of many such actions.
In this proposal we formulate four general themes in this setting which are all strongly related among each other. The first one is the research project of our PostDoc Alvaro Liendo, the second the subject of the thesis of a PhD-student and the remaining ones are joint projects of the principal investigator Hanspeter Kraft with Gerald Schwarz (Brandeis University) and Nolan Wallach (UC San Diego).
- Additive group actions on affine $T$-varieties. In this project we want to study additive groups action on $T$-varieties where $T$ is a torus, assuming some compatibility. This was started by Flenner and Zaidenberg with some very interesting results, and then continued and generalized by Liendo in his thesis.
- Infinite dimensional algebraic groups and varieties. Automorphism groups $Aut(X)$ of certain affine varieties $X$ have the structure of an infinite dimensional algebraic group, a so called ind-group. The most prominent example is the group $GA^n$ of polynomial automorphisms of affine n-space $A^n$. We want to study ind-groups in general, always having in mind the groups $Aut(X)$ and taking $GA^n$ as a guide line.
- Nullcones in representations of reductive groups. The nullcone of a representation $V$ of a reductive group $G$ is the closed set $N_V$ of unstable points, i.e. of those $v$ in $V$ such that the closure of the orbit $Gv$ contains the origin. The two basic question to be answered here are the following.
- For which $V$ the nullcone $N_V$ is reduced? - And what is the number of irreducible components of $N_V$?
- Separating invariants and secant varieties. Separating invariants for a $G$-variety $X$ have been introduced by Derksen and Kemper. It is a subset $S$ of the invariants which separate the same points of $V$ as all invariants together. It is not difficult to see that there are always finite sets of separating invariants, and one can show that the minimal number of them can be bounded by 2n+1 where n is the dimension of the invariant ring. The question is strongly related to the so-called secant variety. Given a closed subset $X$ of $K^N$ we have to understand $X-X$ which is defined as the closure of ${x-x' | x,x' in X}$ in $V$, and to calculate its dimension. If $X$ is a cone this set is usually called the (first) secant variety of $X$.
