# Graduate Level Applied Algebra (Math 6121)

This is a new course created in 2015.

## Nantel Bergeron

tel: 736-2100 Ext 33968
e-mail: bergeron@mathstat.yorku.ca
Office: 2029 TEL 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 (Highly recommended but not required)

## I plan as follow:

I added some reference DF=Dummit and foote(third edition), and S=Sagan(second Edition). 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.
• Introduction Why applied algebra
• Linear Algebra (Recall crash course, Graduate level): [DF Chapter 11] (Sept 7,12,14)
• 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 19, 21, 26)
• Jordan-Holder Theorem (Sept 28,Oct 3)
• Sylow Theorem (Oct 5,10,12)
• Representation of finite groups and characters (over C) (Oct 17,19)
• Midterm (Oct 24): include Linear algebra, Basic groups theory, Jordan-Holder, Sylow Theorem;
Basic Representation Theory: definition of G-module, Representation, invariant subspace (G-submodule), G-morphism.

Lots of my presentation is out of the book The Symmetric Group , Bruce Sagan, Springer GTM 203, (2001). [S Chapter 1]
• Maske's Theorem
• 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
• Preliminary notions in ring [DF Chapter 7,8,9]
• 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)

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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)
• The book Combinatorial Species and Tree-like Structures, Encyclopedia of Math. , Cambridge Univ. Press, (1998), contain many possible Presentation, Project and Applications related to action of groups:
• Polya Theory
• Combinatorial Enumeration
• Species
• and more ...
• The Book Group representations in probability and statistics , Persi Diaconis, Institute of Math. Stat. Lecture Notes, Vol. 11 (1988), contain many possible Presentation, Project and Applications related to representation of groups:
• Discrete Fourrier Transform
• Markov Chain
• Sampling in groups
• and more ...
• The book The Symmetric Group , Bruce Sagan, Springer GTM 203, (2001), contain many possible Presentation, Project 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 ...

## 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)
• 2 Midterms (Writing exams).
• 3 Oral Presentation (Presenting some special topic or long proofs).
• 4 Comprehensive exam (writing the comprehensive exam at the end, Note that for some of you it is one of your Ph. D. requirements).
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%.
Nantel Bergeron
Office: 2029 TEL Building
tel: 416-736-2100 x 33968
email address: bergeron at yorku dot ca
Department of Mathematics and Statistics.
2029 TEL Building
York University
North York, Ontario M3J 1P3, Canada
To Department's Public Page

last revised Sept. 2015