Elena, 

     Your project paper is very good, and the oral presentation was excellent.
   Since you will be able to "recycle" all of this into a first chapter of
   your thesis next year, I have marked a number of small things throughout 
   for you to change or fix, and I have a few general comments:  

   (1) It will help your understanding, I think, to try to provide complete 
       proofs for several of the theorems.  For instance, while the proof 
       of Molien's Theorem (Theorem 4.1) is completely correct, you have 
       not really explained why the trace of the mapping induced by g on 
       S_t equals the product 

              \frac{1}{(1 - \rho_1(g) u)\cdots (1 - \rho_n(g) u)}.     

       Mathematicians often leave out steps like this (out of laziness), 
       but I would say you should be able to write out a full, detailed
       proof and then decide what to leave out.  I know this can seem
       like drudgery if you think you understand what's going on.  But
       it's a bit like practicing scales so that when the time comes 
       to perform a scale passage in a piece of real music, you will 
       have the technique to do something interesting with it(!)

   (2) If you don't want to prove something (e.g. the fact that 
       a matrix of finite order is diagonalizable), in the thesis, 
       you will want to give a reference for that fact.  (But in 
       fact, I think you should also indicate how this is proved because
       the slickest proof is an important application of the criterion
       for diagonalizability in terms of the minimal polynomial of 
       a matrix.  Look this up if you don't recall it from linear
       algebra.)

   (3) Be a bit more careful about hypotheses for the theorems.  Any
       proof that uses the Reynolds operator (e.g. the arguments for Noether's 
       Theorem -- Theorem 3.5, Lemma 4.2, and hence Theorem 4.1 too)
       requires that the characteristic of the field k does not divide
       the order |G|.  In fact, the problem you'll be working on will
       deal exactly with the case where the characteristic of the field
       DOES divide the order |G|.  The finite generation of the invariant
       ring from Theorem 3.5, and Theorem 4.1, are still true in that case, 
       but different proofs are required, and we'll be looking at them
       at the start of the fall.

   Final Project Presentation:  98 (A)

   Final Project Paper: 95 (A)