Graduate Level Applied Algebra (Math 6121) FALL 2023
The FIRST class is SEPTEMBER 7, 2022.
The class is meeting on Tuesday (Vari Hall 1018) and Thursday (Ross S128) from 10:00 to 11:20.
For registered students, more information is available on the eClass page of the class
(follow this link to access eClass),
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%.
Academic Integrity:
All Students are Expected to Engage in Academically Honest Work Academic integrity benefits everyone in our community. It not only helps
you reach the real goal of this class-learning, but also allows for the university and
program to be perceived positively by others. When students are dishonest, they
lose out on valuable learning that will help them perform well in their career. It can
also negatively impact all of the students in the program and at the institution by
creating negative mindsets which may result in fewer outside learning opportunities
for students. Academic dishonesty is any attempt by a student to gain academic
advantage through dishonest means or to assist another student with gaining an
unfair advantage. Academic integrity is important regardless of whether the work
is graded or ungraded, group or individual, written or oral. Dishonest acts are major academic offences and carry serious penalties, ranging from a failing grade on
the plagiarized work to expulsion from the university. For more details,
see York's Academic Honesty Policy and
information on Academic Integrity for Students.
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.
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]
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]
(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 p^{a}q^{b} 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
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