The most famous zeta function is Riemanns. We will discuss its basicproperties, for example its expression as a product over primes due toEuler. Then we consider the analog for a finite connected graph X. Thatmeans we must discuss primes in X. They will be closed paths. Then it iseasy to figure out what the Iharas zeta function of X is. We use thiszeta function to obtain a graph analog of the prime number theoremcounting the number of primes of a certain length in X. When X isregular, the poles of the Ihara zeta function of X satisfy an analog ofthe Riemann hypothesis iff the graph is Ramanujan (meaning that itprovides a good communication network). We will give examples of graphsfor which the Riemann hypothesis is true and other examples for which itis false.

Let $V$ be a representation of a (complex) reductive group $G$. A classical theorem due to Weyl says that the simultaneous invariants (and covariants) of any number of copies of $V$ can be seen in $dim V$ copies. This means that the invariants of more than $dim V$ copies are given by polarization, those for less by restriction. (There are stronger results in case the representation is orthogonal or symplectic.)We are studying the question whether certain geometric objects associated to a representation have a similar behavior. A trivial example is the structure of orbits and their closures which can also be seen in $dim V$ copies. Another example is the rationality of the quotients $V^m/G$ in the sense of geometric invariant theory. Again, it is easy to see that if the quotient is rational for $m$ copies then it is also rational for more than $m$ copies.In the talk we are mainly interested in the set of unstable vectors, the so-called nullcone of the representation. We show that for a certain number $m$, calculated from the weight system of $V$, the number of irreducible components of the nullcone is the same for $geq m$ copies, and that all these components have a nice resolution of singularities. Another interesting question here is if the polarizations define the nullcone, since this has important applications to the computation of invariants. This, in turn, is related to the problem of linear subspaces in the nullcone which is widely open, even in very classical situations.(This is joint work with Nolan Wallach from UCSD. It arose from the study of the representations $C^2otimes C^2otimes cdots otimes C^2$ under $SU_2 imes SU_2 imes cdots imes SU_2$ which play a role in some mathematical aspects of quantum computing.)