Arun Ram: Commutative and Multilinear Algebra 2012Representation Theory 2011

MAST90025
Commutative and Multilinear Algebra

Semester I 2012

Lecturer: Arun Ram, 174 Richard Berry, phone: 8344 6953, email: aram@unimelb.edu.au

Time and Location:
       Lecture Monday 3:15-4:15 Richard Berry 215
       Lecture Wednesday 12:00-1:00 Richard Berry 215
       Lecture/Practical Friday 12:00-1:00 Richard Berry 215

Announcements

No books, notes, calculators, ipods, ipads, phones, etc at the exam.
Tips to avoid freaking out:
- The assignments are designed to take an average of 7 hours per week. This is an average.
Tips for time management:
- It is much easier (and safer) to run 45 min twice each day to attain 28 hours in 4 weeks, than to run for 28 hours solid every fourth week on Sunday.
- To actually run 45 min, it takes me at least 15 min to psyche myself up and convince myself that it is actually not raining and so therefore I should go running, and after a 45 min run I always walk for 5 min and I always go home and have a glass of milk and tell my wife (at length) how cool I am for running 45 min per day. All in all, I waste a good 40 min when I go running for 45 min. If I were more efficient (and every so often, but rarely, I am) then it would only takes me 50 min.
- Measurement of time is a tricky thing and requires real discipline. Teaching and research faculty at University of Melbourne recently had to complete a survey on distribution of their time on the various activities of the job: Do I count the 6 times I had to go check my email and the weather and my iPhone in the time that I spend preparing my Commutative and Multilinear algebra lecture?
Tips for exam preparation:
- The time that a 100m olympic runner (who wins a medal) is actually competing at the olympics is say (5 heats, 7sec each) 40 seconds. Successful performance during these 40 sec is impossible without adequate preparation.
- The time that a Commutative and Multilinear algebra student spends on the final exam is 2 hours. Successful performance during these 2 hours is .....
Consultation hours are the 15 minutes before each class and the 45 min after each class and by appointment. Talk to me before or after class to make an appointment.
Prof. Ram reads email but generally does not respond.
The start of semester pack includes: Housekeeping (pdf file), Plagiarism (pdf file), Plagiarism declaration (pdf file), Academic Misconduct (pdf file), SSLC (pdf file), SSLC responsibilities (pdf file), Beyond third year (pdf file).
It is University Policy that:
“a further component of assessment, oral, written or practical, may be administered by the examiners in any subject at short notice and before the publication of results. Students must therefore ensure that they are able to be in Melbourne at short notice, at any time before the publication of results” (Source: Student Diary).
Students who make arrangements that make them unavailable for examination or further assessment, as outlined above, are therefore not entitled to an alternative opportunity to present for the assessment concerned (i.e. a ‘make-up’ examination).
Printing arrangements from computer Lab G70: Students must use their Unicards to print documents. Locations for Unicard uploaders can be found at: http://www.studentit.unimelb.edu.au/images/facilities/autoloaders.gif

Subject Outline

The handbook entry for this course is at https://handbook.unimelb.edu.au/view/2011/MAST90025. The subject overview that one finds there:

The subject covers aspects of multilinear and commutative algebra as well as two substantial applications. Within multilinear algebra this includes bilinear forms and `multilinear products’ of vector spaces, such as tensor products. Commutative algebra concerns itself with properties of commutative rings, such as polynomial rings and their quotients and to modules over such rings. Both topics have wide application, both to other parts of mathematics and to physics. Much of this theory was developed for applications in geometry and in number theory, and the theorems can be used to cast substantial light on problems from geometry and number theory.

Main Topics

(1) Numbers
(2) Categories and universal objects
(3) Functors: T(V), S(V), Lambda(V), slambda(V), Ext, Tor, E*, and spec
(4) What is a space?
(5) What is Class field theory?
(6) What is Langlands?
(7) What is Yang-Mills?
(8) How to prove the Riemann hypothesis

Assessment

Assessment will be based on up to 60 pages of written assignments (75%: three assignments worth 25% each, due early, mid and late in semester), a and a two-hour written examination (25%, in the examination period).

The plagiarism declaration is available here. The homework assignments are as follows:

Assignment 0: Send me an email indicating if you are enrolled in this course, or not enrolled but auditing, and including any additional information that it may be helpful for me to know. You must do this before 2 March 2012.
Assignment 1: Due 26 March 2012, 5:00pm: Homework assignments should be submitted, by email to aram@unimelb.edu.au.
- Part I: Submit a comprehensive and thorough exposition of best practices for pure mathematics and pure mathematics teaching. Your document must be not more than 15 A4 pages, in 12 point font with at least 1 cm margins. The last page (one page only) should be devoted completely and exclusively to "Suggestions for innovation in undergraduate mathematics teaching" and have this as heading. The remainder of the document should use the following headings in order: (0) Introduction and summary (1) best practices for definitions of mathematical terms, (2) best practices for proofs, (3) best practices for mathematics teaching, (4) best practices for homework assignments in mathematics classes, (5) best practices for mathematics eLearning (6) best practices for mathematics exam writing, (7) best practices for oral mathematics presentations, (8) best practices for mathematical writing, (9) best practices for masters thesis supervision.
- Part II: (a) Make a vocabulary list of all mathematical terms covered in first 4 weeks of the course, (b) provide complete and precise definitions of all mathematical terms covered in first 4 weeks of the course.
Assignment 2: Due 30 April 2012: Submit, by email, to aram@unimelb.edu.au, a document of not more than 20 A4 pages, in 12 point font with at least 1 cm margins which provides a complete and thorough exposition of the definition of the commutative ring of generalized theta functions and generators and relations for the invariant ring and the coinvariant ring for the cases (1) GL₁ (2) GL₂ (3) SL₂ (4) PGL₂ (5) GL₃ (6) SL₃ and (7) PGL₃.
Assignment 3: Due 28 May 2012: Submit, by email, to aram@unimelb.edu.au, a document of not more than 20 A4 pages, in 12 point font with at least 1 cm margins which provides a complete and thorough exposition, including proof, of an analogue of the Chevalley-Shepard-Todd theorem for the commutative ring of generalized theta functions.
The final exam will be 2 hours. No books, notes, calculators, computers, etc. The exam will be as found here (pdf file):

Resources part I: recommended Texts

see http://researchers.ms.unimelb.edu.au/~aram@unimelb/notes.html
Wikipedia
Atiyah and Macdonald, Introduction to commutative algebra, MR0242802
Bourbaki, Algebre, Chapters 1-10 (link to UniMelb library)
Bourbaki, Commutative Algebra, Chapters 1-10 (link to Unimelb Library)
I.G. Macdonald, Algebraic Geometry: Introduction to Schemes, MR0238845
A. Knapp, Introduction to the Langlands program, MR1476501 and S. Gelbart, An elementary introduction to the Langlands program, MR0733692.
P. Deligne, La conjecture de Weil I MR0340258 and La conjecture de Weil II MR0601520

Resources part II: Lectures and lecture notes

Lecture 1, 27 February 2012: Introduction, Housekeeping, Math Grammar, Sets and functions (pdf file)

Math Grammar: Definitions, Theorems and How to do Proofs (pdf file)
Examples of proofs written in proof machine (pdf file)

Lecture 2, 29 February 2012: The rings Z, Z/pZ, Z_p, Z_(p), Z[1/p] -- handwritten lecture notes (pdf file)
Lecture 3, 2 March 2012: Examples of rings; Localization of rings - fractions
Lecture 4, 5 March 2012: The ring of generalized theta functions
Lecture 5, 7 March 2012: Localization of modules, the functor S^-1
Lecture 6, 9 March 2012: Tensor, symmetric, and exterior algebras
Lecture 7, 12 March 2012: Generators and relations
Lecture 8, 14 March 2012: Tensor products, finitely generated abelian groups, and concretization
Lecture 9, 16 March 2012: What is a Space?
Lecture 10, 19 March 2012: Ideals and Spec: prime and maximal ideals
Lecture 11, 21 March 2012: Ideals and Spec: The Zariski topology
Lecture 12, 23 March 2012: Ideals and Spec:affine varieties
Lecture 13, 26 March 2012: Ideals and spec: nilradical and Jacobson radical
Lecture 14, 28 March 2012: Ideals and spec: Nullstellensatz
Lecture 15, 30 March 2012:
Lecture 16, 2 April 2012: Lecture taken by ???: Exterior algebras and determinants
Lecture 17, 4 April 2012: Lecture taken by ???: cofactors, Gram-Schmidt and Smith normal form
Lecture 18, 6 April 2012: Lecture taken by ???: Hilbert series and Grothendieck groups
9-14 April: EASTER BREAK
Lecture 19, 16 April 2012: Lecture taken by ???: Bilinear, sesquilinear and quadratic forms
Lecture 20, 18 April 2012: Lecture taken by ???: Finiteness conditions: Artinian, Noetherian and composition series
Lecture 21, 20 April 2012: Lecture taken by ???: Finiteness conditions: The Jordan-Holder theorem
Lecture 22, 23 April 2012: Lecture taken by ???: Finiteness conditions: Hilbert's Theorem
Lecture 23, 25 April 2012: Lecture taken by ???: Finiteness conditions: Noether normalisation
Lecture 24, 27 April 2012: Lecture taken by ???: Finitenes conditions: integral extensions
Lecture 25, 30 April 2012: Associated primes and primary decomposition
Lecture 26, 2 May 2012:
Lecture 27, 4 May 2012:
Lecture 28, 7 May 2012:
Lecture 29, 9 May 2012:
Lecture 30, 11 May 2012:
Lecture 31, 14 May 2012:
Lecture 32, 16 May 2012:
Lecture 33, 18 May 2012:
Lecture 34, 21 May 2012:
Lecture 35, 23 May 2012:
Lecture 36, 25 May 2012: