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:
Introduction Why applied algebra
Linear Algebra (Recall crash course, Graduate level):
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.
Representation of finite groups and characters (over C)
Lots of my presentation is out of the first chapetr of the book The Symmetric Group , Bruce Sagan, Springer GTM 203, (2001).
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
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)
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
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