# Math 6670, Fall 2017

## Allen Knutson

## Tues/Thurs 11:40-12:55

Algebra is the offer made by the devil to the mathematician. The devil
says: I will give you this powerful machine, it will answer any
question you like. All you need to do is give me your soul: give up
geometry and you will have this marvelous machine.

â€”Sir Michael Atiyah, 2002

The book we'll use for reference is at the bottom
here.
Topics:

Initial notes here.
Next notes.
If you're getting a grade in this class, turn in HW. Due 8/31:

Ex 1.1, 1.2 from those notes
HW due 9/7:
Give the analogue of Taylor's theorem when expanding a
function NN->ZZ as f(d) = \sum_n c_n (d+n choose n).
In particular, show that a polynomial f is integer-valued iff
these c_n are all integer.
What does your analogue give for the non-polynomial function f(d)=2^d?
Prove the Hilbert syzygy theorem for the case of monomial ideals
in x,y.
Let I = < xy-z > and J = < z-t > be ideals in C[x,y,z], where t is
a number. What's the prime decomposition of I+J?
Let I be a homogeneous radical ideal, the intersection of some
minimal prime ideals {P}. Find a formula for the degree of I
(the leading coefficient of the Hilbert polynomial) in terms of
the {P}.
HW due 9/21. Personally, I find the easiest
way to use Macaulay2 to be on my machine from within emacs, but
it *is* possible to
use it online.