Allan Adler has lots of questions about the proof of FLT. I'll accept his
invitation and tackle some of them.

 (1) Mention is made in the proof of the Selmer group of the
 symmetric square representation of the elliptic curve. Naively, I
 would define the Selmer group of a represntation by imitating the
 definition in the case of elliptic curves.  Thus, if rho is a
 representation of Gal on an l-adic vector space V leaving
 invariant a compact and open subgroup L, then Gal acts on V/L and
 for each power of the prime l one has multiplication by that power
 of l as a Gal endomorphism of V/L, with kernel V/L[l^n], if the
 power was the n-th power. That gives us a homomorphism from the
 cohomology group H^1(Gal,V/L[l^n]) into H^1(Gal,V/L)[l^n]. This in
 turn can be composed with the restriction mappings to the
 decomposition groups G_p for all places p of Q and one can define
 the l^n-th Selmer group of rho to be the intersection of the
 kernels of all of these compositions. Of course, for elliptic
 curves, one has the m-th Selmer group for every m, and one would
 like to have that for general l-adic representations also. If one
 has a compatible system of l-adic representations, one can achieve
 that by replacing V/L by the direct sum of all of these V/L one
 gets for all l. But this is pure guesswork. Is it right?

Yes.

A good reference for Selmer groups in the context of compatible
systems of l-adic representations in an article by Ralph
Greenberg, "Iwasawa theory for motives" which appeared in a
book called "L-functions and arithmetic," edited by John Coates
and M. J. Taylor.  It was published as a London Math Society
lecture note volume (#153) in 1991.  Another paper of Greenberg
on the topic is "Iwasawa theory for p-adic representations."
It appeared in the "Iwasawa volume," published in the Advanced
Studies in Pure Math Series (vol #17).  The title of the
volume is "Algebraic number theory -- in honor of K. Iwasawa."
The editors were Coates, Greenberg, Mazur and Satake.

 (2) How does the Selmer group get into the proof of FLT. The sketch of
 Rubin says that one is concerned with showing that the natural mapping
 from the deformation ring R of Galois representations to the deformation
 ring of modular Galois representations is an isomorphism. One is also
 told that one is cutting down to certain representations that satisfy
 additional conditions, hence to quotient rings of R and T and one wants
 to prove that the natural surjection between these quotients is an
 isomorphism. One is told that the Selmer group of the symmetric square
 enters into this. I can imagine how the symmetric square is involved.
 After all, the Zariski tangent space to the deformation ring is
 H^1(Gal,Sym^2(rho)) where rho is the residual Galois representation
 one is trying to lift. But if the symmetric square is involved in this
 way, then it seems one would be trying to prove, e.g. that one is getting
 an isomorphism between the Zariski tangent space of the deformations
 and the Zariski tangent space of the modular deformations. Well, what
 is the Zariksi tangent space of the modular deformations?

There is a lemma which is used.  There is a natural surjection
R --> T, where T is the universal Hecke ring and R is the universal
deformation ring.  In turn, the fact that the mod p represenation
"\rho_0" is modular means that there is a map T-->O hanging around,
where O is a p-adic integer ring.  This latter map gives rise
to an element \eta of T: since T has been shown to be a Gorenstein
O-algebra, the dual to T-->O can be regarded as a map O-->T, and
\eta is the image of 1.  If we view O as a T-algebra, it makes
sense to talk about O/\eta O.  (We are just pushing \eta back
down to O by the original map T-->O.)  On the other hand, let
P be the kernel of the composite map R-->O.  The lemma says that
the structural map R-->T is an isomorphism if the order of P/P^2
is bounded from above by the order of O/\eta O.

The relation between P/P^2 and Selmer groups is "standard" (if
you've read all the papers on deformation theory that have
appeared in the last 7 years).  A reference for a related result
is Proposition 25 of the article by Mazur and Tilouine in
Publ Math IHES volume 71 (1990).

 (3) One is also told that work of Hida, Flach, Kolyvagin, Rubin
 and others is involved. When I look at it, I see e.g. people
 worrying about whether values of l-adic L-functions of the
 symmetric square vanish mod l and many ingenious and interesting
 constructions, but I never see something that unequivocally says
 what the sketch says. I think there is a need for someone to
 provide more details of that part of the sketch with more precise
 pointers to the literature. My feeling, from my own feeble efforts
 to educate myself about this stuff in the last few weeks, is that
 much of what is in the sketch is based on stuff already in print.
 The better one can understand that which is already in print, the
 less mysterious the rest of the proof will become.

I agree that there is a need for a clear explanation of the
new Euler-system techniques which are being used.

 (4) For that matter, I have not seen an unequivocally clear
 description of the two rings Wiles is using in place of R and T.
 I'm told that they are finite Z_l algebras. I can imagine that
 this would come from R and T, in the case of ordinary
 represntations, being both isomorphic to Z_l[[t]], as Mazur proves
 in Lemma 10 of his deformations paper, and that prescribing the
 determinant of a representation amounts to modding out by some
 formal power series, giving us finite Z_l algebras.  Each then has
 only finitely many homomorphisms into Z_l and one way to prove
 isomorphisms between these versions of R and T would beto count
 the homomorphisms to Z_l and show that there are the same
 numbers.  Can one count them and does the count use the stuff I
 mentioned in (3)?

The "counting" takes place in the lemma that I mentioned above.
To prove the upper bound for the order of P/P^2, one uses the
Euler system techniques and the Selmer-group interpretation
of P/P^2 which I mentioned above.  As for the rings R and T,
they are both associated to a deformation problem D.  To define
D, you look at the mod p representation \rho_0 and fix a finite
set of prime numbers Sigma including p and all primes at which
\rho_0 is ramified.  

You look at the restriction of \rho_0 to a decomposition group for
p in the Galois group of Q.  This restriction is assumed to look
like the representation coming from an elliptic curve with
semistable reduction at p.  It may be "flat" (i.e., finite and
flat) but non-ordinary: this would happen in the case of good
supersingular reduction.  It may be "Selmer," meaning like the
representation coming from a good ordinary reduction.  It may be
"strict," meaning like the representation coming from an elliptic
curve with multiplicative reduction but NOT like an elliptic curve
with ordinary reduction.  It may be "ordinary," which means
"strict or Selmer."  You let "dot" be one of "ordinary," "Selmer,"
"strict," "flat" -- but the catch is that \rho_0 has to be of type
"dot".  So you don't have any choice unless \rho_0 is ordinary, in
which case you can let "dot" be "ordinary" or let it be one of
"Selmer," "strict." "Dot" is the first component of the 
deformation problem D.  It means that you look only 
at deformations \rho which, locally at p, are of type "dot."
If I understand correctly, "dot" requires in all cases that,
locally at p, the determinant of \rho differ multiplicatively
from the cyclotomic character by a character of finite order.
This should imply that globally the determinant of \rho is
a character of finite order times the p-adic cyclotomic
character.

Similarly, for each prime q in Sigma which is not p, you look
at the behavior of \rho_0 at q.  If, locally at q, \rho_0 fits
into a standard form that you recognize, you may choose to impose
on the \rho's a local condition at q.

Basically, you look at those deformations \rho of \rho_0
satisfying the chosen local conditions at p and at q's.
There is a universal such deformation, and R is the ring
associated with it.

The definition of T is a little painful.  You have to
perform the exercise of determining what the possible
characters and levels are for weight-two modular forms
whose p-adic representations satisfy the given local
conditions. Roughly speaking, you take the modular curve
X which includes all possible levels and characters, so
X will be intermediate between a modular curve X_1(N) and
the corresponding X_0(N).  The integer N is typically
either prime to p or divisible once by p.  You let H
be the Hecke ring corresponding to X -- you can take
it to be the ring generated by all Hecke operators T_n
in the endomorphism ring of the Jacobian of X.  There
is an eigenform in the picture: this is a cusp form
whose p-adic Galois representation reduces mod p to
\rho_0.  It is convenient to take the coefficients on
this form in a p-adic integer ring (gotten by completing
the ring of integers of the coefficient field of the
form at a prime over p).  Call this p-adic ring O.
The presence of the form leads to a ring homomorphism
H --> O, taking T_n to the nth coefficient of the form.
The maximal ideal of O pulls back to a maximal ideal m
of H.  Now complete H at m and consider the tensor
product (over Z_p) of H_m with O.  That's your Hecke
ring T.

I should have said above the all the deformations to
be considered are O-linear -- you consider only local
O-algebras in making the deformation problem.  So
both T and R are local O-algebras.

 (5) If in fact Mazur proved that the ordinary deformation ring is
 isomorphic to the modular deformation ring in Lemma 10 of his
 deformation paper, what is the gap between that and the
 isomorphism that Wiles proved?

(Adler is talking about page 422 of Mazur's original 
paper on deformations.)  I don't know this subject well
enough to answer.  I suspect that the Mazur proposition
and Wiles's theorem are pretty different in spirit, but
can't add any hard information.  I'll also leave (6),
(7) and (8) for now:

 (6) In the general case of Galois representation and their
 deformations, it is tempting to suppose that if G_S denotes the
 Galois group over Q of the maximal extension unramified outside of
 the finite set S of places of Q, then the set of Galois
 representations of G_S into GL_2(Z_l) should be a p-adic rigid
 analytic space. Is that true?  If so, what is the tangent space to
 that rigid analytic space at a given represntation rho? Would it
 be the cohomology group H^1(G_S,sl_2(Z_l)), where sl_2(Z_l)
 denotes the module of matrices of trace 0, under conjugation by
 G_S via rho? Here, I suppose one would want continuous cochains
 from G_S to sl_2(Z_l), both having their natural nondiscrete
 topologies.  And if true, how does one reconcile that with the
 fact that the Zariski tangent space to the deformation functor is
 H^1(G_S,sl_2(Z/lZ), where G_S acts by conjugation via the residual
 representation to rho?

 (7) The details of the use of the Hilbert irreducibility theorem
 and of the twisted moduli spaces in it are very confusing to me.
 However, I think it is probably one of the most elementary parts
 of the proof. Can someone provide more details of that part, i.e.
 of the proof that if the residual Galois representationat 3
 associated to a semistable elliptic curve E is reducible, then
 there is an elliptic curve E' over Q with irreducible residual
 Galois representation at 3 and the same residual Galois
 representation as that of E at 5?

 (8) Jerry Tunnel's paper proving that octohedral representations
 are modular is quite terse and obviously written for experts. Can
 someone sketch the argument in a more lucid form?

-ken ribet