# Math 6121: Applied Algebra

## Contact information:

Mike Zabrocki
Course will take place TR in Ross S156 from 1-2:30pm
Office: Ross N518
office hours: email me (zabrocki@mathstat.yorku.ca)
I'm available most days but usually not Friday unless you meet me @Fields.

## Course description:

This course covers topics that generally should appear in any graduate algebra course, but are likely to be used in many applications (usually to other areas of mathematics).

Solving linear equations, group theory, polynomial equations, basics of representation theory.

## References:

(most of these references are recommended, but none are required).

Algebra, T. W. Hungerford, GTM Springer.

Abstract Algebra, Dummit and Foote, Willey.

An Introduction to Computational Algebraic Geometry and Commutative Algebra, D. A. Cox, J. Little and D. O'shea, UTM Springer.

The Symmetric Group , Bruce Sagan, Springer GTM 203, (2001).

## Course components:

Students will be evaluated on the following four components. The final grade will be base on the average of the best three.
1. Project (working on an applied algebra project).
2. Midterm exams
3. Oral Presentation - you will be asked to present in class some special topic or long proofs.
4. Final exam - writing the comprehensive exam at the end. Note that for some of you this is one of your Ph.D. requirements.

Participation in class (Being there, asking questions, being curious, etc.) will also be an important component of the class. Because it is difficult to measure and record instances of participation, but I will try and use this as a backup grade component.

You are responsible for completing at least three of these components during the term. Not everyone will feel like they need to do either the project or presentation if they did well on the other components. It will be up to you to schedule a presentation with me or come to me with a proposal for a project.

## 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)
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)

## Schedule:

 Date Topic Notes Sept 8 Introduction, linear algebra notes Sept 13 direct and tensor products, groups, homomorphisms notes Sept 15 G-sets, Burnside's Lemma notes Sept 20 Show off example project, normal subgroup, quotient group notes Sept 22 Isomorphism theorems, Holder's program notes Sept 27 Jordan-Holder Theorem, Sylow Theorems notes Sept 29 Sylow Theorem, if $p~|~|G|$ abelian, $\exists x \in G$ with $order(x)=p$ notes Oct 4 Sylow Theorem, examples and beginning of representation theory notes Oct 6 Representation theory, Maschke's Theorem notes Oct 11 Representation theory, Schur's Lemma notes Oct 13 Representation theory, characters notes Oct 18 Scalar product on characters notes Oct 20 Examples, Fourier transforms, convolution product Oct 25 practice problems for midterm Oct 27 no class - Fall Reading Days Nov 1 Midterm exam Nov 3 Rings and Fields (Ch.7 of Dummit and Foote) notes Nov 8 Rings and Fields (Ch.7/8 of Dummit and Foote) notes, my scanned notes Nov 10 Rings and Fields (Ch.8 of Dummit and Foote) notes, my notes from 11/10 and 11/15 Nov 15 Yohana presents on polynomials rings YS's notes, slides, notes Nov 17 P.I.D.s, U.F.D.s, Noetherian rings, prime/irreducible/maximal ideals notes Nov 22 $F[x]/I$ polynomial algebra. Chinese remainder theorem. notes Nov 24 Grobner bases Sage example, notes Nov 29 Bogdan presents modules, short exact sequences, snake Lemma PB's notes, notes Dec 1 Hilbert series, algebraic geometry and course evaluations

## Presentations:

Choose a Theorem or topic from the list in the course outline and present it in class. Coordinate with me in advance to decide when to present it. I will allocate a 45 minutes of in class time for your presentation. That means that you must choose thoughtfully and carefully what material you will be able to cover in the allotted time. You will be evaluated on the correctness, the amount of material that you are able to cover, as well as the timing and preparation that you put into your presentation.

## Projects:

Take a theorem or a construction that we cover in this class and implement it as a computer program. In order to complete this component there are three parts.

First, you should write a proposal that states the theorem or construction you plan to implement and a description of the main function explaining what is the input and output. You should submit that proposal to me and clear it before you do the next steps. Second, you should write the programs and make sure that each function is well documented (a description of what the function does, a description of the input parameters and output, and a couple of examples of how to use the function). Third, you should should have on a separate page an example of of the use of the functions that you implemented with an explanation of how to read the output.

We will discuss this more in class. Here is an example in Sage that I prepared about the function which converts a linear transformation to a matrix (and back again) (and the text version). This is an implementation of the results that we discussed the first day of class. I also presented a proposal for this (and I want you to do the same) that is discussed in the class notes from Sept 13.

You may use any computer language that you wish.