Question: I wish to use one of your course notes as the text for my course. Is it OK for each of my students to download it and print it out?
Answer: Yes, that's OK.
Question: I wish to hand out copies of the chapter on free groups from your Group Theory notes to my topology class. Is this OK?
Answer: Yes, provided you don't charge for them.
Question: Can you reformat your notes so that I can read them on my six-inch Kindle?
Answer: I did this with the three most popular sets of notes. It's easy to reformat them --- all you need to do is remove the margins and change the print area and fonts to roughly double the number of pages. However, then many tables and math displays don't fit, and it takes to time to fix them.
Question: Can you produce an epub or azw version of the notes?
Answer: No. There are programs that translate TeX into epub, but they don't work very well, and the result requires a lot of manual editing. There are many complaints about the mathematics in the books Amazon sells.
Question: Do you plan to publish the notes as books.
Answer: I had planned to complete some of the notes and publish them as books, but because of the rampant piracy, there seems to be little point in doing this.
I do plan to put some of the notes on Deep Blue where they will remain permanently available, with a permanent URL.
Question: What if you get run over by a big Mack truck?
Answer: I won't care. That's the only good thing about getting run over by a big Mack truck --- you no longer care.
Why are Shimura varieties the "right" objects?
Number theorists have long been interested in modular functions and modular forms, which are functions on the complex upper half plane. Elliptic modular curves are the spaces on which these functions live. An important property of modular curves, which helps explain why they are of interest to number theorists, is that they have canonical models over number fields.
A modular curve is the quotient of a one-dimensional bounded symmetric domain (= hermitian symmetric domain) by a congruence subgroup. Remove the "one-dimensional", and you get a Shimura variety. Shimura varieties are known to have canonical models over number fields, and so have a reasonable claim to being the "right" generalization of modular curves.
From another perspective, a modular curve is a moduli variety for one-dimensional abelian varieties with additional structure. Remove the "one-dimensional" and you get a PEL Shimura variety. I suspect algebraic geometers would have considered these to be the "right" generalization if Shimura hadn't proved that Shimura curves (not of PEL-type) also have canonical models. Shimura varieties with integral weight are moduli variety for motives with additional structure, at least conjecturally.
Shimura varieties play two roles in the Langlands program, first as sources for Galois representations and second as a test of Langlands's idea that all L-functions arising from algebraic geometry are automorphic. For the first, PEL Shimura varieties suffice. Much is known about the zeta functions of Shimura varieties and almost nothing about the zeta functions of other varieties (except for elliptic curves over the rational numbers and modest generalizations).
Thus, Shimura varieties are certainly good objects for number theorists to study (but not the only good objects).
Question: Does the Hodge conjecture have any applications? (Majid Hadian, November 4, 2013).
Answer: Most Shimura varieties (those of abelian type
with rational weight) are known to be moduli varieties for
abelian motives with Hodge cycle structure. Thanks to
Deligne's theorem on Hodge cycles on abelian varieties, this
makes sense over any field of characteristic zero, and can
be used to realize the canonical models as moduli varieties.
If one knew the Hodge conjecture, this description would
persist into characteristic p and would be very helpful in
proving the Langlands-Rapoport conjecture (L&R arrived at
the statement of their conjecture by assuming that the
Shimura variety mod p is a moduli variety for motives). This
is all discussed in my second article at the Seattle motives
conference (1991/1994).
More specifically, L&R prove their conjecture in their paper
for Shimura varieties of PEL type assuming (a) the Hodge
conjecture for CM-varieties; (b) the Tate conjecture for
abelian varieties over finite fields; (c) the Hodge standard
conjecture for abelian varieties over finite fields. I
proved that (a) implies (b) and (c), so all one needs is (a)
to make their proof work word for word. (This is where I'm
permanently stuck, but the conjectural theory of rational Tate classes is
designed to get around (a).)
Once one has the L-R conjecture, the main obstruction to
proving Langlands's conjecture that the zeta function of the
Shimura variety is automorphic was always the fundamental
lemma, but this has now vanished.