The set of unstable vectors of a representation $V$ of a reductive
group $G$, the so-called {\\it nullcone\\/} $N_V$, contains a lot of information about the geometry of the representation $V$. E.g. if $N_V$ contains finitely many orbits then this holds for every fiber of the quotient morphism $\\pi_V\\colon V \\to V/\\!\\!/ G$.

The Hilbert-Mumford criterion allows to describe the nullcone as a union
$\\bigcup G V_\\lambda$, using maximal unstable subspaces of $V_\\lambda
\\subset V$ annihilated by a 1-parameter subgroup $\\lambda$ of $G$. They
correspond to maximal unstable subsets of weights which allows some
interesting combinatorics.

We will give some methods how to determine the irreducible components
$GV_\\lambda$ of the nullcone and will describe their behavior if one
considers several copies of a given representation $V$. A rather complete
picture is obtained for the so-called $\\theta$ representations studied by
Kostant-Rallis and Vinberg. E.g. we were able to show that for the 4-qubits
$Q_4:={\\bf C}^2\\otimes {\\bf C}^2\\otimes {\\bf C}^2 \\otimes {\\bf C}^2$ the
nullcone has four irreducible components all of dimension 12 for one copy
and 12 irreducible components for $k\\geq 2$ copies. These 12 components
decompose into 3 orbits under the obvious action of $S_4$ on $Q_4$, each one
consisting of 4 elements, of dimensions $8k+4$, $8k+3$ and $8k+1$.

(This is joint work with Nolan Wallach.)

We will show how the combinatorics of symmetric functions can give many
new and classic results in the theory of permutation enumeration. Among
other things, a well known result of Garsia and Gessel will be derivied
through this perspective.
This is the second of two talks on the subject; however, this talk is almost
self contained. For those not attending the first talk, we only assume
familiarity with the elementary and homogeneous symmetric functions and the
notions of descents and inversions for permutations.

Physicists have long studied spectra (or eigenvalues) of Schroedinger
operators and random matrices, thanks to the implications for quantum
mechanics. Analogously number theorists and geometers have investigated
the spectra of the differential operators known as Laplacians associated
to certain surfaces with a Riemannian distance. For surfaces with
symmetries coming from number theory, this has been termed \"arithmetic