Announcements:
(Sept 8, 2016) I wil continue to update this web page adding in specific
details about the course over the next week or so. Announcements about
the course will be available here.
(Sept 8, 2016) John Campbell provided notes for the first day. I've posted
them (and will continue to post them) in the schedule section below.
(Sept 19, 2016) As promised, I posted an example of some Sage code in the
Project section of this page. As we cover more
material give some thought about how to turn a theorem into a program.
(Sept 21, 2016) I've updated the project to get the documentation better.
If you want to see how the exercises are done, here are the solutions
of your classmates for
1-11
2
4
7
8
10
11.
(Oct 2, 2016)
I was missing a detail in the proof that if $G$ is a $p$-group
and $H$ a proper subgroup, show that $H \neq N_G(H)$.
(Oct 4, 2016) Rather than cover the proof of Sylow's theorem in full
detail I handed out
an outline (updated: Oct 6)
that includes the proofs with spartan explanations and some details left as exercises.
Fill in the proofs and complete the exercises. Note: the version that I have here
has a few typo's corrected from the one that I handed out in class. Please let
me know if you find additional corrections.
(Oct 4, 2016) I had on the screen
the color coded picture of symmetric group $S_4$ organized
so that one could 'see' the composition factors that is due to
Franco Saliola.
One calculation that we left in class said that there were
either 1 or 3 subgroups of order 8 in the symmetric group $S_4$. I wasn't able to
easily eliminate a possibility.
Sage is able to tell us the answer quickly:
sage: [H.order() for H in SymmetricGroup(4).subgroups()]
[1,
2,
2,
2,
2,
2,
2,
2,
2,
2,
3,
3,
3,
3,
4,
4,
4,
4,
4,
4,
4,
6,
6,
6,
6,
8,
8,
8,
12,
24]
Therefore it is impossible to have a composition series of $S_4$ whose composition
factors are $\simeq ({\mathbb Z}_2, {\mathbb Z}_2, {\mathbb Z}_3)$.
(Oct 7, 2016) As I mentioned in class yesterday, one idea for the project is
to use group theory to solve the rubix cube (starting with the $2 \times 2 \times 2$
and working your way up to $3 \times 3 \times 3$). Another project might be
to solve the
15-puzzle.
(Oct 7, 2016) An idea for a presentation could be an application of algebra
that we are unlikely to get to in class. I found this list online that was
produced in 1998 by
David Joyner,
updated in 2006 and then dropped off the internet in 2016.
The subtitle is
Representation theorists
WILL rule the world one day just you wait :-).
(Oct 9, 2016) Its taken me a few days but I have put together a few more
practice problems.
You will have to do things like this on the exam so I suggest you do them, but
I am not going to collect them (I will read or help you with solutions on
request).
(Oct 15, 2016) For the presentation of representation theory I have been
most closely following
The Symmetric Group:
Representations, Combinatorial Algorithms, and Symmetric Functions
by Bruce Sagan
(Oct 18, 2016) Here is a link to a
visualization of
the discrete Fourier Transform. The discrete Fourier transform
is the
decomposition of data into cyclic group irreducibles, so you can imagine
that the idea of representation theory might also be visualized in the same
way. (
Warning: only view in Chrome or Firefox on a computer that can
handle the visual processing).
(Oct 20, 2016) The example that I took about Fourier transforms,
convolution product, and the probability of walking around the circle
was taken from the book
Group representations in probability and
statistics. Institute
of Mathematical Statistics Lecture Notes-Monograph Series, 11. Hayward,
CA (1988) by Persi Diaconis. Mainly read p. 7 (for convolution and Fourier
transform), 21 (variation distance), 24 (Lemma 1) and
p.25(bottom)-27(top) is where the example is.
(Oct 21, 2016) I am going to put the final exam and comprehensive exam for
this class on Monday, December 5, 2016 from 12-3pm. If you anticipate a conflict, please
let me know as soon as possible.
(Oct 23, 2016) I promised
practice problems
for the midterm exam. Confession: this is the midterm from last year. We can
go over on Tuesday in class *if* you prepare them and want to...if not,
we can always start ring theory. (NOTE: there was an error on problem #2 of the one
that I posted yesterday. I had the definition of the product slightly off.
Make sure that you have the version with the name 102416practicemidterm)
(Oct 23, 2016) John Campbell has made
available his solutions to the exercises (version from 10/23/2016)
that I mentioned in class, hence I post them here.
(Nov 4, 2016) I mentioned in class that for the presentations, please set up
a topic and date with me along with a rough outline of your presentation
(either by email or in person).
You will be evaluated on the quality of the presentation and
whether you are able to keep within the timeline of 45min. You should
take good notes and practice your presentation.
(Nov 10, 2016) I did some examples of rings with Sage on my own computer in class.
I copied the text of
the commands I used in a file in case
you want to try some examples and do something similar.
(Nov 17, 2016) I got the room for the final exam it will be
VH 1016 on Monday, December 12, 12-3pm.
(Nov 29, 2016) You can't google
Snake Lemma without coming across this
proof that appears
in the 1980 film
It's My Turn.
A good application of the Snake Lemma seems to be more elusive.
(Nov 29, 2016) I did some examples with polynomial rings in Sage and I posted
the
text of the
commands (after I cleaned them up a bit) in a file.
(Dec 3, 2016) I noted at the end of class: I will post practice questions
for the final/comprehensive exam ASAP (probably this weekend). I will be
out of town Dec 6-9 so December 5 is about the only day you have to ask me
questions about the class directly before the final. You can still email me,
but I don't know what my response time will be during the week.
(Dec 3, 2016) I again promised
practice problems that I would post.
These questions were taken directly from the final from last year. I will try to
post a few more practice problems later.
(Dec 4, 2016) Here are a few more
practice problems.
To find more practice, look through the problems in Chapter 1-9 of Dummit and Foote.
(Dec 10, 2016) John sent around an updated
version
of solutions to some practice problems for this course.
Outline:
Introduction Why applied algebra?
Linear Algebra:
Vector spaces and ${\mathbb C}^n$, Linear transformations and matrices.
Direct sums, tensor products, symmetric and exterior tensors.
Group Theory and representation Theory
Groups, morphisms, subgroups, G-sets (and G-morphisms), Isomorphism Theorems and quotient groups.
Jordan-Holder Theorem, Sylow Theorem
Representation of finite groups and characters (over C)
Maske's Theorem
Schur's lemma
Structure of the space of G-endomorphisms
Structure of the inner space of characters on G
Theorem the number of irreducible representations for G is equal to the number of conjugacy classes of G
Preliminary notions in ring
Euclidian domain
Principal ideal domain
Unique Factorization domain
Polynomial rings
Grobner basis with emphasis on algorithmic aspect and computational geometry
solving polynomial system of equations
(with some application to robotics and computational geometry)