# Graduate Level Applied Algebra (Math 6121) FALL 2022

The FIRST class is SEPTEMBER 8, 2022.
The class is meeting on Tuesday and Thursday from 10:00 to 11:20 in Ross South 128.

## Nantel Bergeron

e-mail: bergeron (at) yorku (dot) ca
Office: 2029 Dahdaleh Building

## Books

Algebra, T. W. Hungerford, GTM Springer. (recommended but not required)
Abstract Algebra, Dummit and Foote, Willey. (Highly recommended but not required)
An Introduction to Computational Algebraic Geometry and Commutative Algebra, D. A. Cox, J. Little and D. O'shea, UTM Springer (recommended but not required)
The Symmetric Group , Bruce Sagan, Springer GTM 203 (recommended but not required)

## Evaluation:

Students will be evaluated on five aspects (which are parts of the life of any living mathematician). The final grade will be base on the average of the best three.
• 1 Project/Homework (working on an extended project or working on exercises) [to be emailed to instructor regularly]
• 2 Midterms (Writing exams). [this will be online, with zoom. Each student will have its own zoom room]
• 3 Oral Presentation (Presenting some special topic or long proofs). [This would be a zoom presentation to the class]
• 4 Comprehensive exam (writing the comprehensive exam at the end, Note that for some of you it is one of your Ph. D. requirements). [we plan to have the comprehensive exam "in-person", with strick distancing and protection. If this is impossible for some (with justification), we will follow the online protocol. ]
5 Participation in class (Being there, asking questions, being curious, etc.) is ALSO an important aspect of the evaluation. It may help increase any of your average above by up to 10%.

## I plan as follow:

Reference DF=Dummit and foote, and S=Sagan. This could be useful if you want to work on some exercises. Just pick some in those chapter.
Remember that it is important to work the math in order to learn it.

Midterm (Oct 20): include Linear algebra, Basic groups theory, Jordan-Holder, Sylow Theorem;
Basic Representation Theory: definition of G-module, Representation, invariant subspace (G-submodule), G-morphism.
• Introduction Why applied algebra
• Linear Algebra (Recall crash course, Graduate level): [DF Chapter 11] (Sept)
• THM For any fin. Gen. vector space V (over C)
• V has a (ordered) basis B
• dim(V) = |B| =n is well defined
• L: V ----> C^n where L(v)=[v] is an isomorphism
• THM For any linear transformation T: V ---> W, and fixed bases in V and W, There is a unique matrix M=[T] such that LoT=MoL
• All questions about T can be answered using algorithms on matrix [T]
• End(V) = Mat(nxn) and Aut(V)=Gl(n)
• Direct sums and tensor products have corresponding operations on bases and linear transformations.
• Group Theory and representation Theory [DF Chapter 1,2,3,4,5,6]
• Recall : Groups, morphisms, subgroups, G-sets (and G-morphisms), Isomorphism Theorems and quotient groups. (Sept)
• Jordan-Holder Theorem (Sept - Oct)
• Sylow Theorem (Oct)
• Representation of finite groups and characters (over C) (Oct)
Lots of my presentation is out of the book The Symmetric Group , Bruce Sagan, Springer GTM 203, (2001). [S Chapter 1]
• Schur's lemma
• Structure of the space of G-endomorphisms
• Structure of the inner space of characters on G
• THM the number of irreducible representations for G equal the number of conjugacy classes of G
MIDTERM (tentative date: Oct 20) .
• Preliminary notions in ring [DF Chapter 7,8,9] (Nov)
• 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)
• Modules over a ring (Advanced linear algebra) [DF Chapter 10,12]
• Chinese Remainder Theorem

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--- If time allows, Module over PID... it is cover in more details in Math 6122.
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• Modules over PID (Advanced linear algebra) [DF Chapter 10,12]
• Chinese Remainder Theorem
• Classification of finitely generated modules over PID
• Classification of finitely generated abelian groups
• rational canonical form
• Jordan canonical form

## Proposed Presentation, Project and Applications:

My list is not inclusive and is just suggestions
• Pick a Theorem above and present it in class (let me know in advance to coordinate when to present it)
Some suggestion from books:
• An Introduction to Computational Algebraic Geometry and Commutative Algebra, D. A. Cox, J. Little and D. O'shea, UTM Springer.
Presents several algebraic tools that are useful for geometry and related applications.
• Division algorithm for multivariates polynomials and its properties
• Hilbert's finite basis theorem
• Aspects of Grobner Basis theory.
• how to use Grobner basis to solve robotic problems.
• Combinatorial Species and Tree-like Structures, Encyclopedia of Math. , Cambridge Univ. Press, (1998),
Contains projects and applications related to action of groups useful for counting problems:
• Polya Theory
• Combinatorial Enumeration
• Species
• Group representations in probability and statistics , Persi Diaconis, Institute of Math. Stat. Lecture Notes, Vol. 11 (1988),
Contains pojects and applications related to representation of groups useful for probability and statistics:
• Discrete Fourrier Transform
• Markov Chain
• Sampling in groups
• The Symmetric Group , Bruce Sagan, Springer GTM 203, (2001),
Contains projects and applications related representation theory and the symmetric group:
• program the Young Natural Representation
• Present symmetric functions
• Introduce Robinson-Schensted algorithm and its consequences (What was the original application of Schensted?)
• and more ...
• Representation Theory of Finite Groups: an Introductory Approach , Benjamin Steinberg, Springer 2016,
This is a great other source for representation theory and what we can do with it:
• proof of Burnside's paqb solvability theorem

Nantel Bergeron
Office: 2029 Dahdaleh Building
tel: 416-736-5250
email address: bergeron at yorku dot ca
Department of Mathematics and Statistics.
2029 Dahdaleh Building
York University
North York, Ontario M3J 1P3, Canada
To Department's Public Page

last revised Aug 2022